UC-NRLF 


B   M   2SD   M3b 


AN 


ELEMENTARY  TREATISE 


THEORY  OF  EQUATIONS 


SAMUEL   MARX    BARTON,    Ph.D. 

PUOFESSOR   OF   MATHEMATICS.    UNIVF.KSITY    OF   THE   SOUTH 


BOSTON,   U.S.A. 
D.   C.    HEATH   &  CO.,   PUBLISHERS 

1899 


A 


"B 


Copyright,  1899, 
By  D.  C.  heath   &  CO. 


CAJORl. 


.T.  S.  Gushing  &  Co.  -  Berwick  &  Smith 
Norwood  Mass.  U.S.A. 


PREFACE. 

Ix  this  treatise  it  is  my  aim  to  give  the  elements  of  Deter- 
minants and  the  Theory  of  Equations  in  a  form  suitable,  botli 
in  amount  and  quality  of  matter,  for  use  in  the  undergraduate 
courses  in  our  colleges  and  universities.  To  this  end  I  have 
endeavored  to  make  the  work  in  every  part  readily  intelligible 
to  the  average  student  who  has  become  proficient  in  algebra 
and  the  elements  of  trigonometry.  All  use  of  the  calculus  has 
purposely  been  avoided.  While  the  presentation  of  the  sub- 
ject has  necessarily  been  condensed  to  suit  the  requirements 
of  college  courses,  great  pains  has  been  taken  not  to  sacrifice 
clearness  to  brevity.     It  is  a  short  treatise,  but  not  a  syllabus. 

Part  I  treats  of  Determinants.  The  first  two  chapters  give 
the  fundamental  theorems,  with  examples  for  illustration. 
The  third  chapter  consists*  of  applications  and  special  forms 
of  determinants,  followed  by  a  collection  of  carefully  selected 
examples.  These  three  chapters  on  determinants  should  serve 
as  a  helpful  introduction  to  the  study  of  this  interesting  class 
of  functions. 

Part  II  treats  of  the  Theory  of  Equations  proper.  The 
principal  elementary  theorems  concerning  algebraic  and  nu- 
merical equations  are  deduced.  After  a  brief  introduction, 
giving  definitions,  etc.,  there  follows  a  chapter  on  Complex 
Quantities,  a  subject  which  seems  worthy  of  more  space  than 
is  usually  allotted  to  it  in  so  elementary  a  treatise.  This 
chapter,  however,  is  given  not  so  much  for  use  in  the  chai)ters 
that  follow,  as  with  the  hope  tliat  it  may  prove  useful  to  the 
iii 


911343 


iv  PREFACE. 

student  who  pursues  later  in  liis  course  the  study  of  the  Theory 
of  Functions.  As  all  the  theorems  considered  have  become 
classic,  no  special  references  to  authors  consulted  seem  neces- 
sary in  the  body  of  the  book.  After  Chapter  IV  I  have  fol- 
lowed quite  closely  Burnside  and  Panton,  though  in  some 
places  the  general  arrangement  has  been  altered  to  make  the 
necessary  abridgments  Avhile  securing  clearness,  and,  wherever 
it  seemed  desirable,  the  method  of  proof  has  been  changed. 
Almost  every  theorem  is  elucidated  by  the  complete  solution 
of  one  or  more  representative  examples.  I  desire  to  call  spe- 
cial attention  to  this  feature  of  the  book,  which  will  surely 
commend  itself  alike  to  teacher  and  pupil.  In  Chapter  XI 
I  have  striven  to  make  the  rather  complicated  process  of  the 
solution  of  numerical  equations  as  simple  as  possible.  It 
would  defeat  the  object  of  this  treatise  were  much  space 
devoted  to  these  methods,  which  are  laborious  and  of  no  great 
practical  value,  but  what  is  given  is  complete  in  itself.  Hor- 
ner's method  is  explained  in  detail. 

The  following  works  have  been  most  helpful  in  the  prepa- 
ration of  the  treatise,  Muir  and  Burnside  and  Panton  in 
particular  furnishing  many  examples:  Baltzer,  Theorie  mid 
Aiiivendung  der  Deterniiuantem,  1881;  Burnside  and  Panton, 
Theory  of  Equations,  1892  ;  Carnoy,  Cours  d'Alg^bre  Supe- 
rieure,  1892;  Hoiiel,  Conrs  de  Calcul  Infinitesimal,  1878; 
Klempt,  Lehrbuch  zur  Einfalirunq  in  die  Moderne  Algebra, 
1880;  Muir,  A  Treatise  on  Determinants,  1882.  Todhunter's 
Theory  of  Equations,  Chrystal's  Algebra,  \o\.  1,  Scott's  Theonj 
of  Determinants,  and  that  excellent  little  American  work  by 
Professor  L.  G.  Weld  {A  Short  Course  in  the  Theory  of  De- 
terminants) should  also  be  mentioned;  and  the  author  has 
consulted  with  profit  the  well-known  works  of  Serret,  Peter- 
sen, Biermann,  Matthiessen,  and  others. 


PREFACE.  V 

The  author  gratefully  acknowledges  his  indebtedness  to 
Dr.  1).  E.  Smith,  of  the  State  Normal  School,  at  Brockport, 
N.Y.,  to  Professor  William  H.  Echols,  of  the  University  of 
Virginia,  who  have  read  the  manuscript  and  made  suggestive 
criticisms,  and  to  Professor  R.  D.  Bohannan,  of  the  Ohio  State 
Universit}^,  and  Dr.  J.  H.  Gore,  of  the  Columbian  University, 
Washington,  who  have  kindly  read  the  revised  proof  sheets, 
and  given  many  valuable  suggestions,  though  he  does  not  wish 
to  hold  them  in  the  least  responsible  for  the  method  followed 
in  the  treatment  of  the  subject,  nor  for  any  errors  that  may 
have  crept  into  the  work. 

SAMUEL  M.  BARTON. 
Sewanek,  Tenn.,  1899. 


TABLE  OF  CONTENTS. 
PART  I. 

DETERMINANTS. 

CHAPTER   I. 

Origin,  Notation,  and  General  Definition  of  Determinants. 

ARTK  LE  l-AOE 

Historical  note 1 

1.  Permutations 2 

2.  Permanences  and  inversions 2 

3.  Cliange  of  class  of  permutations 3 

4.  Number  of  even  and  odd  permutations  in  a  group      ...  4 

5.  First  definition  of  a  determinant 4 

6.  Second  definition  of  a  determinant 5 

7.  General  rule  for  the  expansion  of  a  determinant        ...  6 

8.  Rule  of  signs 0 

9.  Determinants  as  the  result  of  elimination           ....  7 

10.  Interpretations  of  a  determinant  array 8 

11.  Values  of  unknowns  of  two  simultaneous  equations    ...  8 
Examples 0 

12.  Three  simultaneous  equations 10 

13.  Diagram  for  expanding  determinants 11 

Examples 12 

14.  Values  of  x,  y,  and  z  from  three  simultaneous  equations    .        .  12 

15.  Four  simultaneous  equations !•"> 

16.  Definitions  concerning  elements  and  rows  .        .        .         .14 

17.  Other  notations 14 

Examples 16 


Vlll  CONTENTS. 

CHAPTER   II. 
Properties  of  Determinants. 

AKT.  PAGE 

18-21.    Elementary  theorems  relating  to  determinants      ...  18 

Examples 20 

22.  Determinant  minors 2.3 

23.  Development  of  a  determinant 25 

24.  Application  of  these  principles 27 

25-28.    Special  theorems  concerning  determinants     ....  29 

29.  Development  of  a  determinant 30 

30.  Change  of  order  of  a  determinant 31 

31'.    Evaluation  of  determinants 32 

Examples 32 

32.   Laplace's  development 34 

33-35.   Theorems  concerning  the  addition  of  determinants       .        .  38 

Examples 40 

36.  The  product  of  two  determinants 43 

37.  Euler's  theorem 45 

38.  Rectangular  arrays 46 

39.  Reciprocal  determinants 48 

Examples 49 

CHAPTER   III. 
Applications  and  Special  Forms  of  Determinants.- 

AppUcatioiis  of  Determinants. 
Solution  of    simultaneous  linear    equations,  —  number  of 

unknowns  same  as  number  of  equations  ....  50 

Examples 53 

42-43.    Number  of  equations  greater  than  the  number  of  imknowns  55 

44-46.    Homogeneous  linear  equations 58 

Determinants  of  Special  Forms. 

47.  Symmetrical  determinants 63 

48.  Skew  symmetric  and  skew  determinants 65 

Miscellaneous  examples 65 


CONTENTS. 

PART   II. 

THEORY  OF  EQUATIONS. 
INTRODUCTION. 


ART. 


PAGE 


Historical  note 70 

49.  Elementary  principles 77 

50.  Functions  defined 7H 

51.  Equations  defined 79 

52.  Classification  of  equations 80 


CHAPTER  IV. 
Complex  Numbers. 

54.  Definitions  of  an  imaginary  number 82 

55.  The  complex  number 82 

56.  Successive  powers  of  i 82 

57-61.    Elementary  theorems  concerning  the  complex  number          .  83 

62.    Conjugate  imaginaries 85 

-i-SS.   Theorem  concerning  conjugate  roots 85 

64.    Definition  of  norm  and  modulus 85 

65-66.    Theorems  concerning  moduli 86 

67.    Graphic  representation — Argand's  diagi-am       ....  87 

68-69.    Exponential  form  of  x  +  iy 88 

70.  DeMoivre's  theorem 90 

1 

71.  Values  of                             (e'S)'' 92 

72.  Solution  of  the  equation    x"  -  1  =  0 92 

73.  Solution  of  the  equation     a;"  +  1  =  0 93 

74.  Addition  of  complex  numbers 94 

75.  Subtraction 95 

76.  Multiplication  and  divi.siou '•'0 


CONTENTS. 


CHAPTER   V. 
Properties  of  Polynomials. 

ART. 

77.  Reduction  to  the  form /(x)  =  0 

78.  Theorem  relating  to  polynomials  when  the  variable  receives 

large  values 

79.  Similar  theorem  when  the  variable  receives  small  values 

80.  Derived  functions.      Change  of  form  of  a  polynomial 

spending  to  an  increase  or  decrease  of  the  variable 

81.  Continuity  of  a  rational  integral  function 

82.  The  remainder  theorem 

83.  Tabulation  of  functions 

84.  Graphic  representation  of  a  polynomial 
Examples 


PAGE 

97 


98 
100 

101 
102 
103 
106 
107 
109 


89. 


CHAPTER   VI. 

General  Properties  of  Equations. 

Theorems  relating  to  the  existence  of  a  root  in  special  cases  112 

.  114 

.  115 

.  116 

.  117 

.  118 

.  119 


Existence  of  a  root.     Imaginary  roots 

Theorem  concerning  the  number  of  roots  of  an  equation 

Examples 

Equal  roots 

Imaginary  roots  occur  in  pairs 

Descartes'  Rule  of  signs 


Examples 122 

CHAPTER  VII. 

Relations  between  Eoots  and  Coefficients.  —  Symmetric 
Functions. 

94.  Eelations  between  the  roots  and  coefficients  of  an  equation      .  124 

95.  Applications  of  the  preceding  theorem 125 

Examples 120 

90.    Derived  functions 128 

97.    Multiple  routs, — theorem  .......  129 


CONTENTS.  Xi 

ART.  PAOP, 

98.  Determination  of  multiple  roots ]2".t 

Examples l;50 

99.  Theorem  relating  to  the  passage  of  the  variable  through  a  root 

of  the  equation l;Jl 

100.  The  cube  roots  of  unity 132 

101.  Symmetric  functions  of  the  roots i;33 

Examples 134 


102. 
103. 

104. 
105. 
100. 

107. 

108. 
109. 
110. 
111. 


112. 
113. 
114. 
115. 

IIG. 


CHAPTER   VIII. 
Traksformation  of  Equations. 
Roots  with  signs  changed           .... 
Roots  multiplied  by  a  given  number 
Examples 


.  13G 

.  137 

.  138 

Reciprocal  roots 139 

.  139 

.  140 

.  141 

.  143 

.  144 


Conditions  for  infinite  roots 

Reciprocal,  or  recun-ing,  equations  .... 

Examples 

Roots  diminished  or  increased  by  a  constant  difference 

Examples 

Removal  of  terms 140 

Algebraic  solution  of  the  cubic  equation 147 

Application  to  numerical  equations 149 

Algebraic  solution  of  the  biquadratic  equation          .        .        .150 
Examples 152 

CHAPTER   IX. 
Limits  of  the  Roots  of  an  Equation. 
Definition  of  limits    .... 
Limits  of  roots,  Prop.  I.    . 
Limits  of  roots.  Prop.  II. 
A  third  method  of  getting  the  limits 

Examples 

Inferior  limits  and  limits  of  the  negative  roots 
Examples 


153 
153 
154 
150 
150 
158 
159 


Xll  CONTENTS. 

Separation  of  the  Roots  of  Equations. 

ART.  PAGE 

117.  Separation  of  the  roots 159 

118.  Sturm's  theorem 160 

119.  Application  of  Sturm's  theorem 163 

Examples 164 


CHAPTER  X. 
Elimination, 

120.  Review  of  methods  of  Chapter  III. 

121.  Resultant  by  simple  elimination 

122.  Euler's  method  of  elimination   , 
128.  Sylvester's  dialytic  method 
124.  Other  methods  of  elimination    . 

•  Examples 


166 
1C6 
1G7 
169 
171 
172 


CHAPTER   XI. 
Solution  of  Numerical  Equations. 

125.  Difference  between  algebraic  and  numerical  equations      .        .175 

126.  Theorem  concerning  commensurable  real  roots  oif(x)—0      .     175 

127.  Integral  roots  determined  by  trial 176 

128.  Newton's  method  of  divisors 177 

129.  Application  of  the  method  of  divisors        .        .         ...        .178 

.  178 
.  181 
.  183 
.  184 
.  187 


Examples 

131.  Newton's  method  of  approximation  .    -    . 

132.  Horner's  method  of  solving  numerical  equations 
Examples 

133.  Principle  of  tlie  trial  divisor 

Examples 188 

134.  Negative  roots 191 

Examples 191 

Miscellaneous  examples 191 


Appendix 


197 


THEORY    OF    EQUATIONS. 


PART   I.  — DETERMINANTS. 


CHAPTER   I. 

THE   ORIGIN,    NOTATION,    AND    GENERAL    DEFINITION 
OF   DETERMINANTS. 

As  an  introduction  to  tlie  Theory  of  Equations,  it  seems 
proper  that  Ave  should  devote  a  few  chapters  to  the  discussion 
of  the  important  class  of   functions  known  as  determinants. 

Historical  Note.  The  first  notion  of  Determinants  we  owe  to  Lt'il)nitz, 
who  iu  l(i93  had  observed  the  peculiarity  of  the  expressions  which  arise  from 
tlie  solution  of  linear  equations. 

These  functions  were  first  called  "determinants"  by  Cauchy,  this  name 
being  adopted  by  him  from  the  writings  of  Gauss,  who  had  applied  it  to 
certain  special  classes  of  the.se  functions ;  namely,  the  discriminants  of  binary 
and  ternary  quadratic  forms.  After  Leibnitz  no  furtlier  advance  in,  the  sub- 
ject was  made  until  Cramer,  iu  1750,  was  led  to  the  study  of  such  functions 
in  connecti(m  with  the  analysis  of  curves.  During  the  latter  part  of  tlie 
eighteenth  century,  tlie  subject  was  further  enlarged  by  the  labors  of  Bezout, 
Laplace,  Vandermonde,  and  Lagrange.  In  the  present  century  the  first  mathe- 
maticians who  were  prominent  in  developing  this  branch  of  mathematics  were 
Gauss  and  Cauchy,  and  the  subject  was  also  studied  by  Binet  in  France  and 
Wronski  in  Italy.  We  are  indebted  to  Cauchy  for  the  first  formal  treatise  on 
the  subject.  A  great  impetus  was  given  to  tlie  study  of  these  functions  by 
the  writings  of  Jacobi  in  Vrellc's  Jouriial  in  1841.  Among  more  recent  writers 
who  liave  advanced  the  subject  may  be  mentioned  Hermite,  Hesse,  Joachims- 
tlial,  Cayley,  Sylvester,  and  Salmon. 

Text-books  on  Determinants  were  written  by  Spottiswoode  (1851),  Brioschi 
(1854),  Baltzer  (18,-)7),  Giinter  (1875),  Do.stor  (1877),  Baraniechi  (1871»),  Scott 
(1880),  Muir  (1882),  Weld  (lS«t:3),  and  others.* 

*  The  general  text-books  on  Higher  Algebra  that  devote  n  chapter  or  two  to  Determi- 
nants are  numerous.    Some  of  these  are  referred  to  in  the  author's  rreface.    Gordon's  Vorlt- 
1 


!  *'2' ::>.:/•*•.  •.•'       theory  of  EqUATIONS.  Art.  l 

1.   Permutations.     Let  there  be  a  group  of  elements 
a,  b,  c,  d,  e,  •••, 


represented  by  different  letters  or  the  same  letter  affected 
with  indices  arranged  in  order  of  increasing  magnitude.  If 
we  assemble  these  elements  by  placing  them  in  any  order, 
the  group  thus  obtained  is  called  a  permHtcUion. 

It  is  proved  in  algebra  that  the  number  of  permutations  of 
the  members  of  a  group  of  n  things  is 

1  •  2  •  3  •  •  •  >i,  or  M  !  * 

If  the  members  of  a  group  are  arranged  alphabetically,  or, 
when  represented  by  a  single  letter,  if  the  indices  of  that 
letter  occur  in  order  of  increasing  magnitude,  tliey  are  said  to 
be  written  in  the  natural  order. 

In  one,  and  in  only  one,  of  the  permutations  of  the  members 
of  a  group,  the  members  are  arranged  in  their  natural  order. 
In  every  other  permutation  the  natural  order  is  more  or  less 
deranged. 

2.  Any  two  members  of  a  group  arranged  in  their  natural 
order  constitute  a  permanence.     Thus,  the  pairs 

ab,     ac,     be,     bd,     12,     13,     23, 
are  permanences. 

Any  two  members  of  a  group  arranged  in  an  order  which 
is  the  reverse  of  the  natural  order  constitute  an  inversion. 
Thus  the  permutation 

eadcfb 
presents  eight  inversions, 

ea,     ed,     ec,     eh,     dc,     db,     cb,     fb ; 

sungen  vher  Immrianfentheorie  I.  Bd.  mipht  be  mentioned  in  tliis  connection.  For 
extended  bibliographical  notice,  see  Gunther's  Lehrhtich  der  />eterm>ii(infe>i-T/ieoiie, 
pp.  208  and  200,  Muir's  Theorij  of  DeierminatifK,  and  Scott's  Theory  of  Detetmi.nants. 
*  The  symbol  \v_,  read  "factorial  «,"  is  also  used  to  denote  the  product  of  the  first  n 
whole  numbers,  but  in  printed  work  n  I  is  the  most  convenient  symb<»l. 


Art.  3       GENERAL    DEFIXITION   OF  DKTEliMIXAyrs.         3 

the  permutation  a-,aia^a^a2 

presents  seven  inversions  of  siibscri})ts, 

51,     54,     53,    52,     43,    42,    32. 

The  permutations  of  the  members  of  a  group  are  diviikMl 
into  two  classes,  the  even  or  positive  permutations,  and  tlie  odd 
or  negative  permutations.  Even  permutations  are  those  which 
contain  an  even  number  of  inversions.  Odd  permutations  are 
those  which  contain  an  odd  number  of  inversions. 

The  permutations 

deabc,    32541 

are  even  (positive),  because  each  contains  an  even  number  (six) 
of  inversions ;  while  the  permutations 

daehc,     32451, 

are  odd  (negative),  because  each  contains  an  odd  number  (five) 
of  inversions. 

3.  Theorem.  A  permutation  changes  its  class,  from  even  to 
odd  or  from  odd  to  even,  token  an//  tiro  of  its  members  are  infi'i- 
changed.  /^q    - 

Let  a,  b,  c,  ■■■,q  be  the  indices  of  tlie  elements  of  a  certain 
permutation.     Now  form  the  product 

F^(b-a)(c-a)(d-a)    .     .     .     .     ('/-«)■ 
(c  -  b){d  -b)    .     .     .     .     ('I  -  b) 
(d-  c)    .     .     .     .     (7-  '•) 

■Uj-P) 
of  the  differences  two  and  two  of  these  indices,  taken  by  sul> 
tracting  each  of  them  from  all  those  which  come  after  it  in  this 
permutation.  To  each  inversion  there  will  correspond  a  nega- 
tive difference ;  therefore,  P  will  be  positive  or  negative  accord- 
ing as  the  permutation  belongs  to  the  even  or  to  the  odd  class. 
This  assumed,  let  g,  k  be  the  indices  of  two  rb-mcnts  whitdi 


4  THEORY   OF  EQUATIONS.  Art.  3 

are  to  be  interchanged  ;  the  product  P,  relative  to  the  original 
permutation,  can  be  put  under  the  form 

1  2  3  4 


P=  ±  l(6-a)(c-a)-..  X  (g-a)(g-b)-'-  x  {k-a){k-b)---  x  (k-g), 

the  group  (1)  embracing  all  the  factors  not  contained  in  the 
groups  (2),  (3),  (4). 

If  we  interchange  g  and  k,  the  group  of  factors  (1)  undergoes 
no  change ;  the  groups  (2)  and  (3)  will  only  be  interchanged, 
the  one  for  the  other ;  the  factor  (4)  alone  will  change  its  sign, 
and,  therefore,  the  permutation  will  change  its  class,  which 
was  to  be  proved. 

The  sign  of  the  product  P,  which  determines  whether  the 
permutation  is  even  or  odd,  is  called,  for  brevity,  the  sign  of 
the  permutation,  and  hence  the  name  positive  and  negative  is 
given  to  the  even  class  and  odd  class  respectively. 

4.  TiiKOKEM.  Of  all  possible  penmitations  of  the  members 
of  a  group,  one-half  are  even  and  one-half  are  odd. 

Supjiose  all  the  possible  permutations  written  down.  Now, 
let  a  new  set  of  permutations  be  formed  by  fixing  upon  any 
two  of  the  members  and  interchanging  them  in  each  permu- 
tation. The  even  permutations  will  thus  be  clianged  to  odd, 
and  the  odd  to  even.  That  is,  for  every  even  permutation  in 
the  old  set  there  is  an  odd  one  in  tlie  new,  and  vice  versa. 
But,  as  is  evident,  the  new  set  of  permutations  is  the  same  as 
the  old,  only  differently  arranged.  Hence  in  either  set  there 
are  as  many  even  as  there  are  odd  permutations,  or  one-half 
the  permutations  are  even  and  oncvhalf  are  odd. 

5.  Definition.  ^Ve  can  now  give  our  lirst  definition  of  a 
dotcn-minant.  A  determinant  of  order  n  is  the  algebraic  sum 
of  the  permutations  of  a  product  of  >i  elements 

aib.jc^  •••  k„_il„ 

obtained  either  by  the  interchange  o^  letters,  or  by  the  inter- 
change of  subscripts^.  Z^''"'    '    ^'     ' 


If 


Art.  G      GENERAL   DEFIXITIOy    OF  UETEUMIXAyrs.  ,j 

[We  must  of  course  remember  that  permutations  of  the  even 
class  have  the  +  sign ;  those  of  the  odd  class,  the  —  sign.] 

The  function  aib.,  —  a.,bi  of  the  four  quantities 
a„     bi, 
«2,     b.,, 
is  obtained  by  assigning  to  a  and  b  written  in  alpliabetical 
order,  the  suffixes  1,  2,  and  2,  1,  corresponding  to  the  two  per- 
mutations of  the  numbers  1,  2  (th.e  second  term  being  minus, 
because  2,  1  is  odd) ;  and  adding  the  two  products  so  formed. 

Similarly  the  function 

chh<^3  —  "1^2  +  (iA<-\  —  ^QhCi  4-  a-Ac,  —  aJj.^Ci  (1) 

of  the  nine  quantities 

cii,     bi,     Cj, 

€12,        bo,        Cg, 
C'S)       ^3)       ^3; 

is  obtained  by  adding  algebraically  all  the  products  abc  which 
can  be  formed  by  assigning  to  the  letters  (retained  in  their 
alphabetical  order)  suffixes  corresponding  to  all  the  permutur 
tions  of  the  numbers  1,  2,  3. 

In  like  manner,  we  could  form  a  similar  function  of  the  4th 
order,  of  the  sixteen  quantities 


«1, 

K 

Cl, 

du 

a,, 

h, 

Co, 

d,, 

«3, 

K 

c„ 

(h, 

a^, 

K 

Ci, 

ch- 

These  functions  are  Determinants  according  to  our  first  defini- 
tion. In  these  functions,  the  quantities  a^  bi,  c,,  dy,  a^  etc., 
are  called  elements,  or  constituents. 

6.  Second  Definition.  We  see  from  the  foregoing  that  a 
determinant  embraces  a  square  number  of  elements,  and  this 
leads  us  to  a  notation  —  a  square  array  of  the  elements  between 
two  vertical  lines,  thus : 


THEORY  OF  EQUATIONS. 


Art.  6 


A  = 


ctj     b.,    C2 

ttg         63         C3 


(1) 


and  we  give  as  our  second  definition,  embodying  this  notation, — 
fa"  determinant  of  a  system  of  n^  elements,  which  are  arranged 
/  in  n  rows  of  n  elements  each,  or  n  columns  of  n  elements  each, 
is  the  algebraic  sum  of  alL_possible  products  of  n  of  these 
elements,  no  two  of  which  belong  to  one  row  or  to  one  column, 
the  sign  of  any  product  being  +,  if  the  term  is  an  even  per- 
mutation;  — ,  if  the  term  is  an  odd  permutation. 

7.  It  follows  at  once  from  this  definition  that  a  general  rule 
for  the  expansion  of  a  determinant  array  is : 

Write  down  all  the  products  which  can  be  formed  by  taking 
as  factors  one,  and  only  one,  element  from  each  column  and 
each  row  of  the  array.  Of  these  products,  the  number  of 
which  is  n ! ,  one  half  involve  the  even  permutations,  and  the 
other  half  involve  the  odd  permutations  of  the  subscripts 
1,2,3,  ...n. 

Now  give  to  those  products  the  2^osHive  sign,  if  the  permuta- 
tions of  the  subscripts  are  even;  the  negative  sign,  if  the  per- 
mutations are  odd,  and  take  their  algebraic  sum.  The  result 
is  the  expanded  form  of  the  determinant  array.  This  method 
of  expansion  is,  however,  of  little  practical  value. 

8.  Rule  of  Signs.  The  diagonal  aib-yC^  •••  l„  is  called  the 
principal  diagonal  of  the  determinant.  The  product  afi^c^  •••  /„ 
of  the  letters  in  the  principal  diagonal  always  has  the  sign  +, 
because  in  this  permutation  the  letters  as  well  as  the  suffixes 
occur  in  the  natural  order. 

[      To  determine  the  sign  of  any  other  product :  first,  put  its 

I  letters   in   alphabetical   order ;    then   count   the   interchanges 

necessary  to  bring  the  subscripts  into  the  order,  1,  2,  3,  4,  -••, 

'  of  the  svibscripts  in  the  principal  diagonal.     If  they  make  an 


Art.  9      GENERAL    DEFINITION   OF  DETEUMl S AS  I. >.  7 

even  number,  the  term  is  affected  with  +  ;  if  they  nuike  an 
odd  number,  the  term  is  affected  with  — . 

A  better  rule  is :  To  determine  the  sign  of  any  term,  count  it.f 
number  of  inversions,  making  the  sign  j)lus  or  minus  according  a^ 
that  number  is  even  or  odd. 


EXAMPLES. 

1.  What  sign  is  to  be  attached  to  the  term  aj)-cjli\f^f.,  in 
the  determinant  of  the  seventh  order  ? 

Here  four  interchanges  put  the  subscript  1  first,  tlien  five 
interchanges  put  2  in  second  i)lace,  then  three  put  4  in  fourth 
pLace,  then  two  put  5  in  fifth  place,  and  finally  one  inter- 
change puts  6  between  5  and  7 ;  hence  in  all  there  are  fifteen 
interchanges,  consequently  the  sign  of  the  term  is  — .  Or, 
more  simply,  the  number  of  inversions  is  15,  an  odd  number, 
therefore  the  sign  of  the  term  is  minus. 

2.  In  a  determinant  of  the  fourth  order,  find  the  signs  of 
the  iQVTCi^-.^a^iCid^;  aib^Csd.,;  ctjbf'id^;  a^)jp-/li. 

3.  In  a  determinant  of  the  fifth  order,  find  the  signs  of  the 
terms  :  a-jb^Cod^ei ;  aJyoC^die-j ;  biaod^c^e^ ;  eia-^dib^Ci- 

9.  Determinants  most  frequently  occur  as  the  result  of 
elimination  from  linear  equations.  For  example,  solving  the 
two  simultaneous  linear  equations, 

ciiX  +  &i.y  =  miy 
ttoX  -\-b-^  =  m2, 

we  readily  get    *  x  =  —^;= j-^ (1) 

O162  —  t'/'i 

1  a,m.,  —  a..mi  ,,,-. 

and  y  =  -^ '-r (-) 

ttiWo  —  api 


8  rriEORY  OF  EQUATIONS.  Art.  9 

Tlie  common  denominator  of  these  two  values  of  x  and  y 
is  a  function  of  the  coefficients  of  x  and  y  in  the  given  simul- 
taneous equations.     This  function 

a,b.,  -  a  A  ■     ■ (3) 

is  the  determinant  of  the  coefficients  ctj,  bi,  cu,  h.^,  and  is  com- 
monly expressed  by  the  symbol 


Cly      hi 

iU        bo 


(^) 


This  symbol  is  called  a  determinant  array,  and  the  quantities 
a  I,  hi,  a.2,  b-i  are  called  elements.     (Art.  5.) 

The  polynomial  (o)  is  called  the  e.vpansion  of  the  determi- 
nant. Since  each  term  of  this  expansion  is  the  product  of  tico 
elements,  the  deternnnant  is  said  to  be  of  the  second  order. 

10.  From  our  general  theory,  as  given  in  Arts.  G,  7,  and  8, 
we  recognize  the  equation 


Oo    bo 


ai&2  —  ^2^1 


as  an  identity ;  but,  in  practice,  we  find  more  suitable  rules 
for  expanding  determinants,  and  these  we  shall  examine  later. 
The  first  member  of  the  above  identity  is  thus  interpreted : 
The  determinant  array  of  the  second  order  must  be  under- 
stood to  mean  that  the  product  of  the  elements  on  the  diagonal 
]iassing  from  the  lower  left-hand  corner  to  the  upper  right- 
hand  corner  of  the  array  is  to  be  subtracted  from  the  product 
of  the  elements  of  the  other  diagonal. 

Here  and  elsewhere  we  use  the  sign  =  to  denote  identity. 

11.  The  numerators  of  the  fractions  in  equations  (1)  and 
(2)  of  Art.  9  may  also  be  written  in  the  form  of  determinant 
arrays.  Thus  the  values  of  x  and  y  in  the  given  simultaneous 
equations  are 


Art.  11       GENERAL   DEFINITION  OF  DETKIiMINANTS. 


m^ 

bi 

tiu 

b. 

Cly 

bi 

a. 

b.. 

1.  Expand 

I  b     a 

1     -2 

2.  Evaluate 

4         3 

3.  Evaluate 

4.  Evaluate 

5.  Evaluate 

6.  Expand  and  reduce 

7.  Expand  and  reduce 

8.  Expand  and  reduce 

9.  Expand  and  reduce 


and   y  = 


EXAMPLES. 


a. 

7H, 

Oi 

Vli 

a  I 

b. 

«2 

b. 

Ans.  a-  —  h-. 


Ans.    11. 


0     - 

11 

9         0  1' 

1     4 

0     0 

100     50 

50 

25 

cos  X  sni X 

sin?/  Qosy 

—  1  sin  a  I 

sin  «  —  1    I 

1  —  tan  X 
tan  X         1 
\  x  +  y    x-y 


Ans.  cos  {x  4-  y). 


X  +  y     X  —  y  I 

Solve,  by  Arts.  9  and  11,  the  following  simultaneous  c<iua- 
tions  : 

10.  G .<• -I- 5 // =  40,  10.c  +  3.v  =  <;r). 

11.  2 .7;  4-  7  ?/  =  52,  3  .r  -  5  //  =10. 

12.  2x-ly=    8,  4.v-9.r=19. 
15  4 


13.   ^5  =  ..,   -: 

X     y  X 


=  4. 


10  TIIEOUY  OF  EQUATIONS.  Art.  12 

14.    ax  —  hy  =  c,   ex  +  cuj  =  b. 

12.   Again,  solving  the  three  simultaneous  linear  equations, 
ciiX  +  bjj/  +  CiZ  =  nil, 


we  obtain 


a.x  +  boij  4-  '%2  =  m.,, 
a^x  +  bs!/  +  C3Z  =  Ms, 


and 


afi^Cs  —  aiftgCa  +  a263Ci  —  a-JbiC^  -\-  a-JjiCo  —  a3^2^i 
afi.m^  —  a^b^nii  +  ci.b^mi  —  a.^^m^  +  fi^bim.,  —  ajhm^ 


(1) 
(2) 
(3) 


O162C3  —  (fi^aCo  +  a^^gCi  —  a2'->iC3  +  «3^i<'"2  —  '^^s'-'i^l 

The  common  denominator  of  these  three  fractions,  which 
express  the  values  of  x,  y,  and  z,  is  the  determinant 

«!        ^1        Ci 

ct2    b,    C2        (4) 

«:,     b.,     c. 


The  function 

ciibXs  —  aiZ>oC2  -|-  02^3^1  —  ciobiCs  +  tts&iCo  —  O3&2C1  • 


0"5) 


is  called  the  expansion  of  the  determinant  (4),  and  since  each 
term  of  this  expansion  is  the  product  of  three  elements,  the 
determinant  is  said  to^be  of  the  third  order. 

Note.  Such  examples  as  the  one  given  here  are  simply  to  sliow  that  flo- 
terminants  often  occur  as  the  result  of  elimination.  The  reader  will  learn  in 
Ciiapter  III  that  the  process  of  elimiuatiou  is  much  simplified  by  the  use  of 
determinants. 

13.  Since  the  determinant  (4)  is  identically  equal  to  the 
function  (5),  we  have,  arranging  the  terms  of  (5)  in  a'  epn- 
venient  order. 


Art.  13      GENERAL   DEFINITION    OF  UKTEUMIXANTS.      H 


«1 

^'1 

Ci 

«2 

i>2 

c., 

«3 

h 

^'3 

If  a  line  be  drawn  tlirough  each  triad  of  letters  forinin<?  a 
term,  we  have  the  following  diagrams,  which  furnish  an  excel- 
lent device  for  assisting  the  memory  in  expanding  a  deter- 
minant of  the  third  order,  viz.,  for  the  positive  terms : 


Snd  ternu^^ 

\ 

\ 

„  \ 

^>      1 

Srd  term. 

\''V 

'^      ^%      ^X 

and  for  the  negative  terms 


6lh  term'' 
5th  term/ 
Ulh  term/ 


X.  / 


In  making  practical  use  of  these  diagrams,  it  is  customary 
to  carry  out  the  multiplication  as  each  stroke  is  made.  No 
similar  diagram  exists  for  a  determinant  of  higher  order  than 
the  third. 


12 


THEORY  OF  EQUATIONS. 


Art.  13 


EXAMPLES. 

Expand  by  tliis  method  the  following  determinants ; 


X     y     z 

V     to     u 

t      r      s 

{. 

X     -2z     -2/2 

—  y     —2x        z"^ 

-z          2y     —x^ 

Evalnate  the  determinants, 

. 

4         5         2 

-1         2     -3 

6-4         5 

f_ 

4     -1     -2 

0         3         0 

3     -7 

4 

9  8  7 
6  5  4 
3    2    1 


10. 


a 

-2a 

6 

36 

-c       4d 

2c 

3  J   -4  6 

a     a 

a 

6     a 

6 

c      c 

a 

2    0 

3 

1     3 

5 

2    6 

10 

15 

4     -3  1 

2 

10         5 

0 

3         7 

-1 

0    4 

2 

3    5 

4 

6    0 

14.  The  numerators  of  the  fractions  in  Art.  12  may  also  be 
written  in  the  form  of  determinant  arrays,  and  thus  we  have 
for  the  values  of  x,  y,  and  z  in  the  given  simnltaneous  equa- 
tions : 


7Jll 

hi 

c, 

«! 

Wl 

Ci 

Wi2 

bo 

c. 

tto 

V12 

c, 

m^ 

h. 

C3 

y  = 

(h 

m^ 

C-l 

a, 

h, 

C] 

Cli 

hi 

Ci 

a,. 

K 

Cl' 

02 

b. 

c. 

«;j 

h. 

c-i 

^3 

h. 

C3 

Art.  15     GENERAL   DEFINITION   OF  DETFUMINANTS.      Vi 


and 


«1 

fh 

Ml 

a.. 

h 

vu 

Ch 

h 

tih 

Cll 

61 

Cy 

«2 

h 

c. 

«3 

b. 

Cs 

Note.  It  may  be  observed  that  the  numerator  of  the  fraction  ex- 
pressing the  value  of  x  may  be  furmeil  from  tlie  denominator  of  the 
same  fraction  by  replacing  rti,  n.y,  03,  the  coefficients  of  x,  by  the  absolute 
terms  mi,  m-2,  J/13,  respectively.     Similarly  for  y  and  z. 


15.   The  solution  of  the  four  simultaneous  equations 

UiX  +  bill  +  CyZ  +  cliiv  =  ?Ui 
a^  +  b.,)/  +  c.jZ  +  (Iw  —  m2 
ci^x  +  b.^i  +  c^z  +  f's?^  =  ^'^3 
UiX  +  Z>4.?/  +  C42;  +  d^^o  =  vii 


(1) 


would  shoAv  that  the  values  of  x,  y,  z,  and  w  are  expressed  by 
fractions  having  a  common  denominator  which  is  a  function  of 


the  sixteen  coefficients  Oi,  &i,  Cj,  d„  a^ 
is  a  determinant  of  the  fourth  order, 
identical  equation : 


•  ••,  etc.     This  function 
We  have  tlie  following 


ttl 

&i 

Ci 

f?, 

tto 

h 

C2 

^2 

«3 

h 

C3 

C?3 

a* 

64 

C4 

ch 

=  (iibx./li  —  a^h^^d^  —  cifb^c^i  +  afiiC/l^ 
+  aihfiiil2  —  afiiC-fh  —  aJ)iC./h  +  fhh''A'h 
+  a^JiC^di  —  UibiC/ls  —  (t^b^c/h  +  ciibic/h 
+  cQ)^Cidi  —  aJjiCtd.^  —  o.jMidi  -f  a^ijCifi's 
+  ajbiCyd^  —  ctibsCido  —  ap-jp^y  -]-  a.j64<y?, 

+  «./^2^4f?l  —  «4&2C3'^^  -  "3*4C/^  +  «4/>3''/^ 


(2) 


The  solution  of  five  simultaneous  linear  equations  involving 
five  unknoAvn  quantities  would  give  rise  to  a  determinant  of  the 
fifth  order,  the  expanded  form  of  which  contains  120 (=5!) 
terms. 


14 


THE  our   OF  EQUATIONS. 


Alt.  15 


tto  ^2  f"2  ^^2  ^2 
<-h  h  ("3  C?3  ^3 
(f4        64        C4        CZ^        64 


f?-. 


Oi^2<^3f?4^5  ±  etc. 


(3) 


Similarly,  the  solution  of  n  sinmltaueous  linear  equations 
involving  n  unknown  quantities  would  give  rise  to  a  determi- 
nant of  the  ?jth  order,  the  expanded  form  of  which  contains 
n !  terms.     The  determinant  array  may  be  written  thus : 


(4) 


ttl 

h 

Cl      • 

•    h 

02 

b. 

C2   • 

•  h 

Kg 

h 

Ci      • 

'  k 

16.  It  is  evident  from  the  foregoing  that  the  determinant  of 
the  nth  order  involves  nr  elements,  which  agrees  with  Art.  6. 

The  horizontal  ranks  of  elements  are  called  rows  of  the 
determinant,  and  the  vertical  ranks  are  called  columns.  The 
rows  are  numbered  from  the  top  row  downward,  and  the  col- 
umns from  the  left-hand  column  to  the  right.  A  line  is  either 
a  row  or  a  column. 

In  any  determinant,  the  diagonal  from  the  upper  left-hand 
corner  to  the  lower  right-hand  corner  is  called  the  priitcipcd 
diagonal,  as  we  have  had  occasion  to  remark,  and  the  other 
is  called  the  secondary  diagonal  The  terms  of  the  expansion, 
which  are  the  products  of  the  elements  on  these  diagonals,  are 
called  respectively  the  principal  term  ^.nd  the  secondary  term. 
Thus  in  the  determinant  (4),  of  the  preceding  article,  aih-f.^  •••  ?„ 
is  the  principal  term,  o„ft„_iC„_2  •••  Zj  is  the  secondary  term. 

17.  Another  notation  for  the  determinant  of  the  Jitli  order 
is  the  following : 


Art.  17      GENERAL   DEFINITION   OF  DETEUMIN ANTS.      15 


«! 

«1 

a/" 

•••     a/"> 

Oo' 

a.r 

a. J" 

•••     a,,<"> 

O3' 

Ch" 

(I3" 

...     a3<'" 

«,.'    «„ 


(1) 


iu  which  the  number  of  the  nm  is  indicated  by  the  subscript, 
and  the  number  of  the  column  by  the  sujyerscriiJt. 

Another  notation,  and  one  that  is  very  much  used,  is  the 
following : 


(2) 


a„ 

"12 

«13       • 

•        «ln 

«21 

a.r.2 

«03        • 

•     as,. 

«31 

Ch2 

«33        • 

•       «3,. 

Here  the  number  of  the  roiv  is  indicated  by  the  Jirst  of  the 
two  subscripts,  and  the  number  of  the  column  by  the  second. 
Thus,  the  element  a^  of  the  above  array  is  in  the  third  row, 
and  the  fifth  column. 

There  are  several  simpler  methods  for  writing  determinants, 
when  it  is  perfectly  well  understood  what  the  elements  of  the 
determinants  are. 

Thus,  if  A  denotes  the  determinant  (4)  of  Art.  15,  it  may, 
for  brevity,  be  represented  in  the  following  ways : 

A  =  (a,V3---Q (3) 

A  =  \aAcs-h\ (-4) 

that  is,  simply  by  placing  the  principal  term  within  brackets. 
The  notation  2  ±  a,V3  •••  ^..  is  also  used  to  represent  A;  this 
expressing  its  constitution  as  consisting  of  the  sum  of  a  num- 
ber of  terms  (with  their  proper  signs)  formed  by  taking  all 
possible  permutations  of  the  n  suffixes.  With  this  notation 
determinant  (2)  of  this  article  would  be  expressed  by 

2  ±  «n«2-/'33---« (^) 


16 


TUEOllY  OF  EQUATIONS. 


Art.  17 


EXAMPLES. 
1.   Expand       o-j    ?/i     1 


^2    1/2    1 


2.   Evaluate 


3.   Evaluate 


4.    Evaluate 


5.    Expand 


^ns.   — 15. 


3 

2     4 

7 

G    1 

5 

3    8 

1     4 


6 
-3 
^ns.  20 +  8 +  96 -(-60) 


5     0 
4     2 


0-(-12) 


lOtr- 


1  a 
a  1 
6     c 


In  the  following  examples  express  tlie  values  of  x,  y,  and  z 
in  the  notation  of  deterniinauts,  as  in  Art.  14,  and  then  evaluate 
these  determinants  by  the  method  of  Art.  13. 

6.    Solve  the  simultaneous  equations, 
X  +  y  —z  =  l, 
?>x-\-oy  —  Qz  =  l, 
—  Ax-y-\-'Sz  =  \. 


Art.  17      GENERAL   DEFINITION    OF  LETEHMlNANrs.      17 


Here,  by  Art.  14, 

1  1-1 
1  3-6 
1    -1       3 


1 

1 

-1 

8 

1 

-6 

-4 

1 

3 

1 

1 

-1 

8 

3 

-G 

-4 

-1 

3 

y  = 


■^,   !/  = 


1 

1 

8 

1 

-4 

_1 

1 

1 

-1 

8 

-6 

-4 

-1 

3 

3,    z. 


-1  •       -1 

7.  Solve  tlie  simultaneous  equations, 

3x  +  2y  —  4:Z  =  15, 

5a; -3//  + 22  =  28, 

-  ic  +  3  y  +  4  2  =  24. 

Ans.  X  = 

8.  Solve  the  simultaneous  equations, 

4x-'Sy  +  2z=    9, 
■^1  2x  +  5y  —  8z 
6y-2z  +  5x 


2  =  4. 


=  4, 
=  18. 
Alts 


2,y 


3,  2  =  5. 


9.    Solve  the  simultaneous  equations, 
3a; +  2?/+    z  =  23, 
5x  +  2y  +  iz  =  4C), 
10  .K  -H  5  ?/  -f-  4  2  =  75. 

10.    Solve  the  simultaneous  e(iuations, 
2  a; -7// +  42=    0, 
3a; -3//+    2=    0, 
9x  +  oy  +  3z  =  2S. 


CHAPTER    II. 

PROPERTIES   OF  DETERMINANTS. 

From  our  definitions  of  a  determinant,  as  given  in  Articles 
5  and  6,  we  readily  deduce  the  folloAving  important  theorems : 

18.  Theokem.  The  value  of  a  determinant  is  not  changed 
by  substituting  the  columns  for  corresponding  rows  and  the  roivs 
for  corresponding  columns;  that  is, 


«1 

&i 

Ci       ' 

•     h 

«i 

«2 

«3        • 

•     a,. 

02 

b. 

C2     ' 

•     h 

^1 

b. 

bs     • 

•     K 

ttg 

h 

C3     • 

■     h 

~ 

q 

Co 

Cs     • 

•      c„ 

a„ 

K 

c„    • 

••    L 

/i 

h 

Is      ■ 

•      l„ 

YoY,  the  two  determinants  having  the  same  principal  term, 
they  will  be  identical  on  account  of  the  way  in  which  all  the 
other  terms  are  deduced.     (Art.  8.) 

It  follows  that  any  theorem  true  in  regard  to  the  rows  of  a 
determinant  is  also  true  in  regard  to  the  columns,  and  vice 
versa. 

19.  Theorem.  Interchanging  any  tico  roios  (or  columns)  of  a 
determinant,  simply  changes  the  sign  of  the  determinant;  that  is, 


a,   &i   Ci    . 

••    ^1 

a->   b2   C.2    • 

•     h 

h     Cd     Ci     . 

•    h 

ch  bi  Co   • 

•      ^2 

cii   bi   c'l    • 

•    ^x 

&2     ftj     ('2      • 

■    h 

«3    ^3    Cg    • 

•      k 

=  — 

«3   ^3   C3    • 

•      /3 

=  - 

l>3     «3     Cg      • 

•    h 

a„  6„  c„  • 

•      In 

«„   b„  c„    • 

18 

•      In 

^„  a„  o„   • 

•  L 

Art.  21 


PROPERTIES   OF  DETERMINAXTS. 


i:» 


For  this  modification  amounts  to  changing  tlie  index  1  with 
the  index  2,  or  the  letter  a  with  the  letter  b  in  the  different 
terms  of  the  determinant,  and  we  know  that  in  this  case  the 
corresponding  permutation  changes  its  sign,  and  hence  all  the 
terms  of  the  last  two  determinants  will  have  the  same  absolute 
value  as  that  of  the  first,  but  their  signs  will  be  different. 

20.   Theorem.    If  two  rows  (or  cohimns)  of  a  determinant  are 
identical,  the  determinatd  is  equal  to  zero;  thus 


A  = 


«! 

^1 

Ci      • 

•    h 

«r 

K 

^r        • 

.    /, 

a^ 

K 

c,. 

•    /, 

On 

h„ 

c,. 

•           lu 

For,  by  interchanging  the  two  identical  rows,  we  obtain 


Whence 


A  =  -A, 
2A  =  0. 
A  =  0. 


21.  Theorem.     If  each  element  in  any  line  be  mxltiplied  by 
the  name  factor,  the  determinant  is  multiplied  by  that  factor  ;  thus : 


mcii 

b, 

<-'i     ■ 

•         ^1 

ma2 

b. 

C2        • 

•         h 

ma^i 

bs 

C3        • 

•         h 

a, 

^ 

c, 

•  /. 

a.. 

b. 

Co        • 

•  /. 

CJg 

bs 

C3   • 

•  k 

ia„    b„    c„    •••     In 

For  every  term  of  the  determinant  must  contain  one,  and 
only  one,  element  from  any  row  or  any  <-()lumii. 


20 


TIlEOIiV  OF  EQUATIONS 


Art.  21 


Cor.  I.  If  the  elements  in  any  line  are  the  same  multiple 
of  the  corresponding  elements  of  any  other  parallel  line,  the 
determinant  vanishes. 


bi     mbi     1)2 


a  I 

«i 

a. 

«3 

b. 

^1 

h. 

bs 

t'l 

t'l 

C'2 

("3 

rfi 

dy 

(h 

<h 

=  0. 


Cor.  II.  If  the  signs  of  each  element  in  any  line  be  changed, 
the  sign  of  the  determinant  is  changed.  For  this  is  equivalent 
to-multiplying  by  the  factor  —  1. 


EXAMPLES. 
1.    Show  that  the  following  determinant  vanishes ; 

4    3     2     1 

8  8  7  2 
16  2  8  4 
12    6    3    3 

When  the  elements  of  the  first  column  are  divided  by  4, 
they  become  identical  with  those  of  the  last  column. 


f 
5  -^ 

5    1 

4     3 

2    1 


2.  Prove  the  following  identity  : 

2  G     10     2  13 

3  6     15     3  12 
=  G 

2     4       4     3  2     4 

5    2      2    1  5    2 

3.  Show  that  the  following  determinant  vanishes : 
2  0  4  G  1 
13  0  2  7 
2  4  13  2 
4  8  2  6  4 
9     0  5     3     7 


Art.  21 


PR  OP  Eli  TIES    OF  DE  TERM  IN  A  N  TS. 


21 


4.   Prove  the  identity  : 


be  a  a- 
ca  b  b- 
ab     c     c- 


1 

a' 

a' 

1 

b- 

b' 

t 

c- 

c^ 

Represent  the  first  determinant  l)y  A,  and  nuiUiply  the  rows 
by  a,  b,  c,  respectively.     We  have  then 


abc  A  = 


abc 
abc 
abc 


and,  dividing  the  first  column  by  abc,  the  result  follows 
5.    Prove  the  identity 


(3y8 

u 

a' 

a' 

1 

a'     a' 

ySa 

/3 

i8^ 

/3-' 

1 

ft'     /?■' 

8a(3 

y 

r 

/ 

1 

y-'   y 

a^y 

8 

s- 

8' 

1 

S'     8 

6.    Prove 


2  1 

-4  -£ 

6  t 

Prove  the  identity 

1      1  1 

«    ft  y 

«^     ^^  y'^ 


=  (ft-y)(y-  u) («  -  ^)- 


Since  if  3  were  equal  to  y,  two  columns  would  become  iden- 
tical, ft-y  must  be  a  factor  in  the  determinant.  Similarly, 
y -  «  and  u-ft  must  be  factors  in  it.  Hence  the  product  of 
the  three  differences  can  differ  by  a  numerical  factor  only 
from  the  value  oZthe  determinant,  since  both  functions  are  of 


22  THEORY  OF  EQUATIONS.  Art.  21 

the  third  degree  in  a,  (3,  y;  and  by  comparing  the  term  fiy-  we 
observe  that  this  factor  is  -|- 1. 

Note.  If  the  student  is  not  familiar  with  the  application  of  the  so-called 
"  Remainder  Theorem,"  he  might  find  it  to  his  advantage  at  this  point  to  read 
Art.  82. 

Examples  7  and  8  belong  to  a  class  of  functions  called  "  alternating,"  and 
these  particular  determinants  are  known  as  simple  alternants.  The  following 
definitions  may  assist  the  student  (see  Crystal's  Alyebra,  Vol.  I,  Chap.  IV). 

An  integral  function  is  said  to  be  symmetrical  (with  respect  to  all  its  vari- 
ables) when  the  interchange  of  any  pair  whatever  of  its  variables  would  leave 
its  value  unaltered.  For  example,  yz+zx  +  xy  is  a  symmetrical  function  of 
z,  y,  z. 

An  integral  function  is  called  "alternating,"  lohen  the  interchange  of 
any  pair  ivhatever  of  its  variables  changes  the  sign  only  of  the  function.  An 
example  is  (/s  —  v)  (y  —  a)  (a  —  ^) ,  or  the  determinant  of  Ex.  7. 

Now  xhj  +  y^z  +  z^x  is  an  asymmetrical  function ;  that  is,  it  is  not  a  sym- 
metrical function  of  x,  y,  z,  for  the  three  interchanges  x  with  y,  x  with  z,  y 
with  z,  give  respectively 

y^x  +  x^z  +  z^-y, 

z^y  +  y-^x  +  x% 

x'h  +  z'^y-+y^x, 

and,  though  these  are  all  equal  to  each  other,  no  one  of  them  is  equal  to  the 
original  function.  We  observe  from  this  instance  that  asymmetrical  func- 
tions have  a  property,  which  symmetrical  functions  have  not,  of  assuming 
different  values  when  the  variables  are  interchanged :  thus  x'^y  +  y^z  +  z^x  is 
susceptible, of  two  different  values  under  this  treatment,  and  is  called  a  two- 
valued  function.  The  study  of  algebra  from  this  point  of  view  has  developed 
into  a  beautiful  branch  of  modern  algebra,  known  as  the  theory  of  substitu- 
tio)is  (or  t/ie  theory  of  groups)  .* 

8.    Prove  similarly  the  identity 
1111 

I^     ^^     /     I     =-(^-y)(«-S)(y-«)(^-S)(«-^)(y-8). 
u'    (3'    /    8' 

*  See  'Sctto's  Suhsfi(ution/it?ieorie,  Serret's  Cours  DWlgebre  Superieure,  Petersen's 
Algebraiache  Gleichungen. 


Art.  2-2 


PROPEUriES   OF  DETEiniLXAyTS. 


28 


9.    Eeduoe  the  following  detenuinant  to  one  in  which  the 
tivst  row  shall  consist  of  units  : 


9 

4 

10 

5 

0 

1 

4 

3 

7 

2 

5 

5 

3 

0 

1 

4 

Since  20  is  the  least  common  multiple  of  2,  4,  10,  ">,  it  is 
sufficient  to  multiply  the  columns  in  order  by  10,  5,  2,  4 ;  we 
thus  obtain 

20  20    20    20 

0  o      8     12 

70  10     10     20 

30  0      2    IC 


2  •  4  .  10 


Taking  out  the  multiplier  20  from  the  first  row,  10  from  the 
third  row,  and  2  from  the  fourth  row,  we  get  finally 


A  = 


1111 

0     5     8     12 

7     11       2 

15    0    1      8 


10.    Reduce  the  following  determinant  to  one  in  which  the 
firsl  column  shall  consist  of  units : 


A  = 


2    0 

2 

1 

3'\ 

4 

0 

6  ^^6 

7- 

6 

8    4 

4 

5 

22.  Determinant  Minors.  It  is  evident  from  the  notation,  in 
a  s(puire  array,  of  a  deteiminant  of  the  Hth  order,  that  the 
suppression  of  }>  rows  and  of  p  columns  leaves  a  square  con- 


24 


THEORY   OF  EQUATIONS. 


taining  no  more  than  /t  —j)  rows  and  n  —p  columns.  We  thus 
obtain  a  series  of  determinants  of  lower  order,  which  we  call 
minors  of  the  primitive  determinant.     For  example,  in 


«1 

hi 

Ci      • 

•   ^-1 

h 

(U 

b. 

Co         ' 

•    h 

k 

Ch 

Ih 

C3   • 

•     A-3 

k 

let  us  suppress  the  row  and  the  column  which  contain  the 
element  a^ ;  it  will  become  the  determinant  of  the  {n  —  l)th 
order, 


Ay. 


h 

c-i     • 

•     A-, 

h 

h. 

C-i       ' 

•     A3 

h 

K 

f«     • 

•     A-,. 

In 

^±boCs-'-l„ 


which  is  called  the  first  muior  of  A  with  respect  to  the  ele- 
ment Cli- 

As  we  can  repeat  this  operation  on  each  element,  a  deter- 
minant of  the  7<th  order  has  as  ma.ny  Jirst  muiors  as  it  contains 
elements. 

We  designate  these  generally  by  the  large  letters  A,  B,C,'-, 
written  with  the  same  index  as  the  corresponding  element. 
These,  arranged  in  the  order  of  the  elements,  form  the  follow- 
ing table : 


A,    B, 

C\     ' 

•      /M 

Li 

A.    B. 

c,  . 

•    /v:, 

'% 

A„    B„ 

c„   ■ 

•    Jk 

Ln 

ind 

When  we  suppress  any  two  rows  and  two  columns,  the 
remaining  determinant  of  the  (n  —  2)th  order  is  called  the 
second  mUior  of  the  original  determinant. 


Art.  23 


PROPERTIES   OF  DETERMINAXTS. 


By  omitting,  for  example,  the  tirst  two  rows  and  the  lirsl 
two  columns,  Ave  have  the  determinant 


A-4     U 


-~\M' 


_       lO-O^v 


>->^ 


0.^^ 


the  rows  and  tlie  columns  suppressed  have  in  common  the 
elements  of  the  determinant 


«i 


and  the  determinant  of  the  (/i  —  2)th  order  which  precedes  is 
the  second  minor  of  A  with  respect  to  a  determinant  of  t\^ 
second.  pr.d.er.  ^ 

In  general,  by  the  suppression  of  j)  rows  and  of  j)  columns, 
we  get  a  determinant  of  the  {n  —  7))th  order  which  we  call  the 
l^th  minor  of  A  corresponding  to  a  determinant  of  the  ^)th 
"ordeF  formed  by  the  elements  common  to  the  rows  and  col- 
umns suppressed.  The  minor  thus  formed  is  said  to  bex(>^H- 
^>/'eH^e»^a/•^/  to  the  determinant  formed  by  elements  common  to 
the  suppressed  rows  and  columns. 


.# 


v'23.   Development  of  a  determinant  according  to  the  elements  of 
a  row  and  of  a  column. 

Take  the  determinant  of  )(th  order. 


A  = 


«1 

^ 

'h        • 

•    h 

/, 

«2 

h. 

C, 

•    h 

^.' 

«3 

h 

('3        • 

■    h 

hi 

A-..    /.. 


2±o,''/'3---A-„_i^« 


The  different  terms  of  A  which  contain  the  element  ((,  are 
obtained  by  forming  all  the  possible  permutations  of  the  other 


.-J^p 


■J 

26  THEORY  OF  EQUATIONS  Art.  23 

elements,  which  gives  the  deteiiiiinant  of  {n  —  l)th  order, 
2  ±  62C3  •  •  •  ^',,-1^,.,  resu'king/froiu  the  suppression  of  the  first 
row  and  the  first  columiy.  The  determinant  A  will  contain 
then,  first,  a  series  of  ternfs  having  ctj  as  a  factor  represented  by 

ai2  ±  h^'^U  -In- 
Interchanging  a  and  h,  this  becomes 

for  all  the  terms  containing  by)  the  sign  —  is  caused  b}'^  the 
permutation  of  a  and  h.     The  sum  2  designates  the  determi- 
nant obtained  by  omitting  the  first  row  and  the  second  column. 
By  changing  b  into  c,  we  get,  similarly,  the  expression 

which  woidd  represent  the  series  of  terras  having  Cj  as  a  fac- 
tor; the  sum  2  designating  the  determinant  arising  from  the 
suppression  of  the  first  row  and  of  the  third  column,  and  so  on. 
All  the  terms  containing  /j  would  be  represented  by 


(-l)"/,2±«A'"4--^„ 


^^ 


Now,  by  definition,  the  sums  which  accompany  the  elements 
«„  —  6|,  -f  Cj,  •••  (—  1)"/,  are  the  fird^minors  with  respect  to 
these  elements.     Therefore,  we  have  the  following  formula: 

A  =  a,A,  +  b,B,  -K  r.C,  -f  -  +  7.A,     .^.     (1) 


with  the  conditioA  that,  according  to  the  cojnposition  of  the 
determinant,  we  mu^t  atb-ibute  rs  the  minors  Signs  alternately 
positive  and  negative •^"t^^^^^)^  -vv^'^m  o-vaa,  J^-  "K  v^^^^' 
The  number  of  terms  in  the  second  member  of  this  equality 
is  evidently 

n(l  •2.3...y_^J)=  v\, 

the  same  as  the  number  of  terms  in  the  determinant. 


Art.  24  PROPERTIES    OF  DETERMINANTS  27 

Following  the  same  reasoiiiiig,  and  interchanging  successively 
the  indices  two  and  two,  we  arrive  at  the  similar  relation 

^  =  aiA,^a,A,  +  a,A,'T '■■  -h^A,,,   .     .     .     (2) 

■where  we  must  give  to  the  minors  the  signs  alternately  + 
and  — .  This  formula  gives  the  development  of  A  according 
to  the  elements  of  the  first  column. 

It  is  evident  that  there  exists  a  similar  development  for  each 
row  and  for  each  column.  Finally,  to  fix  the  sign  of  the 
minors  in  each  formula,  we  move  the  row,  or  the  column  that 
we  are  considering,  to  the  first  place  by  the  interchanges  of 
the  rows  or  columns,  in  observing  that  the  determinant  changes 
only  its  sign  for  an  odd  number  of  interchanges,  while  it  pre- 
serves the  same  sign  for  an  even  number.     Thus, 

A  =7i2A.,  +  b.B.  4-  c.a  +  •••  +  hLo     ...     (3) 

A  =  a,A,  +  ^^3^3  +  CsC,  +  •  •  •  +  I,Ls ;    .     .     .     (4) 

in  formula  (3)  we  would  alternate  the  signs  commencing  with 
the  sign  —  for  A.2;  for,  to  lead  the  second  row  to  tlie  first 
place,  one  interchange  of  two  rows  suffices ;  in  formula  (4)  it 
is  necessary  to  commence  with  the  sign  +  for  A^,  since  two 
interchanges  of  rows  are  required  to  lead  the  third  row  to  the 
first  place ;  and  so  on. 

24.   Let  us  apply  these  principles  to  a  determinant  of  the 

third  order : 

A  =     ttj     ^1     c'l 

a.,     bo     Co 
0,5     63     c^ 

Expressing  the  minors  with  their  proper  signs,  we  have 

A  =  a^A,  -  &1B1  -\-c,Ci=-  cioAo  -f  ho  B,  -  c.,a = ff,.  I3  -  k'  Ih  +  '-.Oi, 

A  =  a,^li  -(^12-1-%  .13=  -  h,  B,  +  b,B,-b,B,=Ci(-\-i\C,+c,C,, 


28  THEORY  OF  EQUATIONS. 

or,  replacing  the  minors  by  their  vakies, 
A  = 


Art.  24 


a, 

b, 

C2 

^3 

-^ 

a, 

«3 

C2 
C3 

+  c, 

tto 
«3 

&2 
&3 

a.. 

+  h 

C3 

—     Co 

h 
h 

«3 

bi 

C2 

-h 

«2 

Cj 

Co 

+  C3 

«1 

bi 
b2 

«i 

&2 
&3 

C3 

-a. 

&3 

C3 

+  a3 

^2 

Cj 

^\ 

O2 

Co 

+  Z>2 

Cti 

Ci 

-&3 

tti 

^1 

Ch 

C3 

a,, 

(•3 

02 

C2 

Cj 

as 

«3 

—     Co 

«i 

«3 

&1 
h 

+  C3 

62 

By  virtue  of  Avhat  precedes,  we  can  operate  as  follows  to 
ascertain  the  sign  of  a  minor  with  respect  to  any  element. 
For  example,  let  it  be  proposed  to  find  the  sign  of  the  minor  of 
d^  in  a  determinant  of  the  «th  order.  Proceeding  on  the  first 
row  from  cii,  alternating  the  signs  until  we  get  to  the  column 
of  the  cl  elements,  we  reach  c?i  with  the  sign  — ;  we  descend 
tlien  the  column  of  the  d  elements,  changing  the  sign  each  time 
that  we  cross  a  row  until  Ave  arrive,  in  this  manner,  at  d^  with 
the  sign  +  ;  therefore  the  minor  of  d^  ought  to  be  affected 
with  the  positive  sign. 

Again,  let  the  determinant  be  represented  by 


«21 

O12     • 
a22     . 

.     a.2, 

«H 

a*2    • 

■     ■    (iu   ■     ■ 

■     «*, 

a  , 

a...,     . 

.     a 

Art.  26 


FliOPERTIES   OF  DETEliMINANTS. 


29 


and  let  us  seek  the  sign  of  the  minor  relative  to  the  element 
Ua-  To  this  end,  we  must  by  the  interchange  of  rows  and 
columns  lead  this  element  to  the  first  place.  By  ^  —  1  inter- 
changes of  two  consecutive  columns,  the  element  a„  will 
occupy  the  first  position  in  the  ^th  horizontal  line;  then  by 
k  —  1  interchanges  of  rows,  this  element  will  take  the  first 
place  in  the  first  row.  All  these  operations  amount  to  multi- 
plying the  determinant  by  (—  l)*+'-2  or  (—  1)*+'.  The  sign 
of  the  minor  A^i  will  therefore  be  positive  if  the  sum  of  the 
indices  of  the  element  a«  is  even,  and  negative  in  the  con- 
trary case.  It  is  useful  to  observe  that  the  first  minors  of 
the  elements  of  the  diagonal  are  all  positive. 

The  preceding  developments  lead  to  iinportant  consequences 
which  we  shall  now  give,  for  brevity  making  use  of  simple 
determinants  in  illustration. 

25.  Theorem.  When  all  the  elements  of  a  row  or  of  a  col- 
umn are  zero,  the  determinant  is  equal  to  zero.     Thus 

=  0, 


«1 

b,       Ci 

=  0, 

0    &,    c 

«2 

1)2       C.2 

0     b,    c. 

0 

0     0 

0    63    c, 

for  all  the  terms  of  the  development  according  to  the  elements 
of  these  lines  become  zero  by  the  presence  of  the  factor  zero. 

26.  Theorem.  When  each  of  the  elements  of  a  row  (or 
column)  is  zero  except  one  of  them,  the  order  of  the  determinant 
is  lowered  by  unity. 

We  have,  for  example, 


Also 


«,    0 

0 

=  0,^1,  +  0 

.jU  +  O.A,= 

«i 

bt     c. 

a.i    62 

C2 

i,     C3 

«s     63 

Cs- 

0      hi     0 

=  -b, 

a,    c. 

a.2     62    (',, 

"a     fs 

«3        ^3        C3 

30 


THEORY  OF  EQUATIONS. 


Art.  27 


27.  Theorem.  A  determinant  is  reduced  to  its  princijjal 
term  ivheu  each  of  the  eJements  on  one  side  of  the  diagonal  is 
zero.  For,  taking  a  determinant  of  the  fourth  order,  we  have 
successively 

aib^Csd^. 


a, 

0      0 

0 

=  ttj 

&,    0     0 

=  ajb2 

cs    0 

a^ 

b,    0 

0 

h   C3    0 

c,    d. 

Cfg 

h   cs 

0 

b,    Ci     di 

"4 

h    '-i 

<h 

This  appears  at  once  from  equation  (1),  Art.  23,  Avhere  all  the 
terms,  except  the  first,  have  zero  for  a  factor,  and  therefore 
vanish. 

28.  TiiEOKEM.  To  multiply  d  determinant  bi/ ]:>,  it  suffices  to 
'mnltiplij  the  elemevds  of  a  row  or  of  a  column  by  this  factor. 
We  have 

P«I      P^l      l^Cl 

p-  A=  pa^Ai  +  2AB1  +  iX'iC 


29.   If,  in  one  of  the  developments 

«,.4,  +  b,Bi  +  CiC\  +  •••  +  hLi, 

we  replace  the  elements,  Oj,  b^,  Cj  •••  l^,  which  appear  here  by 
those  of  any  other  row,  the  result  is  zero;  the  same  is  the 
case,  if  in  one  of  the  expressions 

rt,yli  +  0*2^42  +  «3^3  +  •••  +  «„-l„, 

we  replace  the  elements  by  those  of  another  column.  For,  in 
substituting,  for  example,  in  the  place  of  the  elements  «i,  &j, 
c'l,  •••  /i,  those  of  the  second  row  a.2,  b.,,  c.^,  •••  l.j,  the  expression 

a,A,  +  b,B,  +  c,C,+  --+hLy, 

represents  the  determinant  obtained  by  this  substitution,  the 
coefficients  A^,  Bi  •••  />,  being  always  the  minors  of  I  he  first 


Alt.  30 


PliOPEHriES   OF  DETEliMl.\AyTS. 


r,\ 


row;    and   this   determiuaut   is    zero,   since    it   contains   two 
identical  rows. 
That  is,  we  have 

Oj.li  +  bjii  +  CiC'i  +  ...  +  /,/.,    =  A, 

a^A,  +  a..  A,  +  a.^A^  +  •••  +  ('„/-/,.  =  A, 

but  ttoAi  +  b.Jii  +  c..Ci  H +  l.,Li   =  0, 

lind  other  similar  relations. 

In  general,  the  expression 

represents  the  determinant  A,  if  J  =  i;  and  is  zero,  if  j  is 
different  from  /. 

Cor.  With  the  notation  witli  two  subscripts,  for  a  determi- 
nant of  the  //til  ortler,  this  property-  is  expressed  thus:  tiie 
developments 

ciijAu  +  o,jA,,  +  •••  +  a„jA„,, 

cij^Aii  +  aj.xl,2  +  -■■  +  cij^A,,,, 

represent  the  determinant  A,  when  _/  is  a  number  of  the  series 
1,  2,  3,  •••  n,  and  equal  to  /;  while,  if^  is  different  from  /,  they 
equal  zero. 


30.  We  can  always  raise  the  order  of  a  determinant  without 
changing  its  value.  Thus,  after  the  preceding  properties,  we 
have  the  equalities 


«! 

^ 

= 

1 

0 

0 

= 

1     X    y      = 

10    0    0 

a^ 

b. 

X 

<'i 

h 

0     a,    h^ 

x    1     0    0 

y 

th 

h 

0     a,    b. 

.'/    t     ih   f>i 

z      II     ((.,    b., 

and  so  on.     The  elements   .r,  ?/,  z,  t,  ii    being   any   quantities 
whatever. 


32 


THEORY  OF  EQUATIONS. 


Art.  31 


31.   Development  and  Evaluation  of  Determinants.     The  fuucla- 
meutal  formula 

A  =  rt,.4i  +  h,B,  +  Cid  +  •••  +  hL, 

enables  us  to  replace  a  determinant  of  the  ni\\  order  by  an 
expression  containing  only  determinants  of  the  {n  —  l)th  order; 
in  this  last  we  can  substitute  for  y1„  Bi,  Cj,  •••  expressions  con- 
taining only  determinants  of  the  (n  —  2)th  order ;  in  continuing 
in  this  way  we  finally  arrive  at  the  value  of  the  determinant  A. 
It  is  necessary  to  give  some  applications  to  indicate  the  steps 
in  the  different  cases  that  may  present  themselves. 

As  we  have  seen,  determinants  of  the  second  order  are  calcu- 
lated directly.     We  have 


«! 


«2        ^2 

3.-4-1.2  = 


a-^h^  —  (uhy. 


14, 


For  a  determinant  of  the  third  order,  of  which  all  the  ele- 
ments are  different  from  zero,  we  would  take,  for  example,  the 


formula 

Uo     bo 
«3     h 


\ 

=  «1 

b. 

Co 

-a. 

b, 

Oi 

+  % 

&i 

Cl 

*3 

bs 

Cs 

bs 

Cs 

b. 

C2 

EXAMPLES. 


=  1 


2      3 


+  3 


-l-2(-2)  +  3(-l)  =  0. 


•'^•l  .'/l   1 

=  X, 

?h 

1 

-X, 

?/l 

1 

+  a-3 

^1 

1 

Xi  y-2  1 

ys 

1 

y?. 

1 

y2 

1 

a^s   Vz    1 

^•i  (y-  -  2/3)  +  x,(ys  -  ?/,)  +  X.,  (?/i  -  ?/o). 


Art.  31 


PROPERTIES   OF  DETERMIXASTS. 


■.V6 


Note.  In  examples  where  certain  elements  are  zero,  we  oufjlit  to  employ 
the  development  aceordinjj  to  the  line  which  contains  the  greatest  number  of 
zero  elements.    Thus : 

=  3. 


10    3 

=  1 

1     1 

+  3    2     1 

2     11 

1     4 

2     1 

2     14 

12  0 
4    10 

13  2 


=  2  I  1     2  I  =  • 
4    1 


14. 


5.   Develop 


6.  Develop 
2  3 
0  1 
0    3 


a 

b 

e 

b 

c 

f 

e 

f 

9 

12    0 
Develop 


b    -c 


0  i- 
-1 


8.    Develop  the  determinants : 
2    1 


3 

5    ()     7 

2     14 


3    0 

2 

1     4 

0 

3    1 

-1 

Ans.  1  -\-  a-  +  b-  +  r. 


0    2    5 

10    4 

, 

3    6    0 

4    0 

0 

2    6 

8 

1    3 

5 

9.   Develop 


1    0 

0 

0 

4    3 

0 

0 

2    7 

8 

0 

5    0 

6 

2 

34  THEOBY  OF  EqUATlONS. 

10.    Develop     0        c  d 


Art.  31 


1         sin  a 
sin  «        1 


32.   Laplace's  Development,  —  Development  of  a  Determinant 
according  to  the  Elements  of   Two  Rows  or  of   Two  Columns. 

Take  the  determinant  of  the  ?;th  order: 


a  I 

^1 

fi     • 

•  h 

Ch 

h 

(-2        • 

•         ^2 

as 

h 

f-s      • 

•        /3 

f'4 

h 

Ci      • 

•        ^4 

n       &„       e,.        ...         /,, 

Let  us  consider  the  principal  term  of  A,  aib.>c./Ii  •••l„;  to  this 
term  there  corresponds  another,  —ciobiCsdi '•'!„,  arising  from 
the  interchange  of  the  letters  a  and  b.  Uniting  these  two 
terms  so  as  to  put  their  common  factor  in  evidence,  we  have 


¥h-l.. 


Let  a  and  b  be  fixed,  and  form  all  the  possible  permutations 
of  the  other  letters,  c,  d,  •••  I,  then  the  terms  of  A,  which  have 
as  a  factor  the  determinant  (ai/>2),  will  be  represented  by 

The  coefficient  of  (ai^j)  is  therefore  the  second  minor  of  A 
obtained  in  suppressing  the  rows  and  columns  which  contain 
these  elements.  We  reach  an  analogous  conclusion  for  the 
coefficients  of  the  determinants  of  the  second  order 

(«A),  (<'A)--("J^„),  («A)  •••("„- A.), 

^^hi(•h  result  from  the  combinations  two  and  two  of  the  ele- 
ments of  the  first  two  columns  of  A.  In  calling  the  second 
minors  Bf,,  Z?,,;,  TJ^,  etc.,  we  find  the  following  development : 


Art  ^2 


PROPERTIES   OF  DETEinfiyANrS. 


8.' 


A  =  (afi2)B,,  +  {aJj,)B,,  +  -  +  (<^,6,.)/i„.  +  {aJ>,)R,.,  +  ... 

+  (<t,.-i/',.)An-.,„. 
The  number  of  terms  of  the  second  member  is  represented  by 


]n(n-l) 


2  .  'J^^ ±1  (1-2.3 


(n  -  2))  =  H  I 


as  it  ought  to  be. 

It  is  important  to  remark  that,  in  the  preceding  formula, 
it  is  necessary  to  attribute  to  the  second  minors  a  sign  in  con- 
formity with  the  vahie  of  A. 

In  the  first  place,  the  second  minor  Bi.,  ought  to  have  the 
positive  sign.  Finally  to  fix  the  sign  of  the  others,  it  is  neces- 
sary by  the  interchange  of  lines,  to  lead  the  coefficients  of  the 
determinant  of  the  second  order  to  the  first  two  places,  in 
preserving  always  the  order  of  the  indices.  Thus,  the  second 
minor  ^13  would  be  negative,  because  one  interchange  of  two 
lines  is  necessary  to  lead  03,  63  to  the  place  of  o^,  b.2 ;  the  minor 
^23  would  be  positive,  for  there  is  necessary  one  interchange 
to  lead  tto,  b.2  to  the  first  row,  and  another  to  lead  (r,,  b-^  lo  the 
second  row.     And  so  on  for  the  others. 

Using  these  principles,  let  us  develop  the  determinant  of 
the  fifth  order: 

cii    bi  Ci  di  €1 

cii     bi  Co  ch  e., 

A  =     ag  — ^  t'a  ds  e^ 

4i     4  Ca  fh  ("i 

«8       ijs       C5       (/.-,       Pi 

according  to  the  elements  of  the  first  two  columns.     It  will 
become,  with  the  abridged  notation  (Art.  17), 

+  (aA)  (c,d,e,)  -  (a/u)  Ovh^s)  +  («A)  (cid,e^)  +  (a^b^)  (ty/^,) 
-  (a3&.5)  (cid-jei)  +  (a A)  (cA^a)- 


36 


THEORY   OF  EQUATIONS. 


Art.  32 


The  same  mode  of  development  exists  relatively  to  any  two 
rows  or  any  two  columns. 

This  development  may  be  applied  to  the  calculation  of  a 
determinant  of  the  fourth  order,  the  expansion  giving  only 
determinants  of  the  second  order.     We  have 


f'l 

b. 

Ci 

rfi 

a^ 

b. 

(h 

d. 

a. 

h 

C3 

(h 

«4 

b. 

C4 

ch 

For  example,  calculate 

3  11 
15  0 
2-2  1 
0  4-5 
Developing,  as  above,  we  have 


A  = 

3  1 
1  5 

1  6 
-5  3 

- 

31 
1  0 

• 

-2  6 
4  3 

+ 

3  2 
1  3 

-2 
4 

1 

-5 

+ 

1  1 

5  0 

2      6 
0      3 

1  2 
5  3 

• 

2       1 
0  -5 

+ 

1  2 
0  3 

• 

2 
0 

-^1 

A  =  14  .  33  +  1  • 

2. 

Calculate 

A  = 

1     2     -1 

2     1         2 

0    3         5 

10         1 

3. 

til     61     0 

0(2    62    0 

as     6,    C3 

a*    bi    C4 

30 +  7. 6-5. 0  +  7- -10 +  3 


398. 


0 

= 

1     2 

. 

5    2 

— 

1 

-1 

3     2 

0 

2     1 

1     2 

2         2 

0     2 

2 

+ 

2     -1 

0     2 

=  -58. 

9 

1 

2 

1     2 

(aA)M.). 


Art.  32 


PEOPERTIES   OF  DETERMIXANT^ 


81 


The  following  example  illustrates  how  the  operation  may  be 
shortened  by  first  bringing  the  zero  elements  into  consecutive 

positions. 


«1  ^1  fl  ^1 

0    b.,  c,  0 

«3   h  ^3  (k 

" 

0    b,  c,  0 

tti  cli  bi  Ci 

0    0    bo.  c. 
«3  f'3  ^3  c,.i 

=  - 

0   0    64  C4 

0 

0      bo 

(■2 

0 

0 

64 

(■4 

Ch 

rfl 

&1 

^"l 

ttg 

CZ3 

63 

C3 

=  -(V4)(«W3)- 


5.    Prove  the  identity 


«1 

bi 

Ci 

Xi 

2/1 

Zi 

tta 

h 

<-'2 

X2 

2/2 

Zo 

h 

1  ^'1 

bi 

('1 

«1        /«! 

71 

0-3 

C.J 

^3 

//3 

^3 

1 

0 

0 

0' 

«1 

/Si 

yi 

=  1  a, 

1     ' 

b-2 

c. 

a,     (3, 

72 

0 

0 

0 

«2 

A 

72 

1  as 

h 

1*3 

«3       /?3 

73 

0 

0 

0 

«3 

fis 

73 

This  appears  by  expanding  the  determinant  in  terms  of  the 
minors  formed  from  the  first  three  columns,  for  it  is  evident 
that  all  these  minors  vanish  (having  at  least  one  row  of  ciphers) 
except  one,  viz.  (a^  b^  Cg). 

In  general  it  appears  in  the  same  way,  that  if  a  determinant 
of  the  2  7?ith  order  contains  in  any  position  a  square  of  «t* 
ciphers,  it  can  be  expressed  as  the  product  of  two  determi- 
nants of  the  mth  order. 

This  is  known  as  Laplace's  Method,*  and  can  readily  be 
extended  to  the  general  case.  Let  any  number  p  of  columns 
be  taken,  and  all  possible  minors  formed  by  taking  p  rows  of 
these  columns.  Each  of  these  minors  is  then  to  be  multiplied 
by  the  complementary  minor,  and  the  deterjuinant  expressed 
as  the  sum  of  all  such  products  with  their  proper  signs. 

*  Pierre  Simon  Laplace  (1T49-1&27),  the  groat  French  mathematician  an.I  astronoiiuT. 


4 


38 


THEORY   OF  EQUATIONS. 


Art.  33 


ADDITION   OF   DETERMINANTS. 

33.  Theorem.  If  every  element  in  any  row  (or  column)  can 
be  resolved  into  the  sum  of  tivo  other^  the  determinant  can  be  re- 
solved into  the  sum  of  tivo  other"!^.    \     " 

Suppose  the  elements  of  the  first  column  to  be  ai  +  a^,  a2+«2j 
«3  +  «3,  etc.  Substituting  these  in  the  expansion  of  Art.  23, 
equation  2,  we  have 

A  =  («i  +  «i)  Ai  +  (€1.2  +  «2)  A  +  («3  +  (h)  A  +  etc. 

=  tti^li  +  a-jA.^  +  a^As  -\ etc.  +  a^Ai  +  UoAo  +  UsA^  +  etc. ; 

or, 
Oi  +  «i     bi     Ci     •■ 
a.,  +  a.,     &2     C.2     '  ■ 


0-3  +  «3        ^3        Ci 


«i  bi  Cj 
a^  bo  c.2 
a,     b.     Co 


4- 


«i     6i     Ci 

«2        ^2        ^2 
«3        ^3       ^3 


which  proves  the  proposition. 
Similarly,  the  determinant 

ch  +  «i     ?>i  +  I3i     Ci 

a.2  +  «2       ^2  +  ^2       f  2 
«3  +  «3        ^3  +  A        C3 

is  equal  to  the  sum  of  the  four  determinants 

{afi^C^  +  («l&2f3)  +  («1^2C3)  +  («]M)- 

In  like  manner  it  follows  that  if  each  of  the  elements  of  one 
column  consists  of  the  algebraical  sum  of  any  number  of  terms, 
the  determinant  can  be  resolved  into  the  sum  of  a  correspond- 
ing number  of  determinants.     Tor  example : 


ai-«i  +  «i' 

b,      C, 

tti 

&i 

Cx 

«1 

^'l 

C\ 

«/ 

^1 

Ci 

O2  — «2+«2' 

b.2      C.2 

= 

a.2 

^2 

c.2 

- 

«2 

f>2 

<^2 

+ 

«2' 

b.2 

c. 

«3-«3+«3' 

h      C3 

Ch 

^3 

C3 

«3 

h 

C3 

"3' 

h 

C3 

^^.r^ 


Art.  35 


PROPERTIES   OF  DKrERMINANTS. 


30 


And,  in  general,  if^one  column  (or  row)  consists  of  the  alge- 
braic sum  of  m  otheili,  a  second  column  (or  row)  of  the  sum  of 
71  others,  a  third  of  the  sum  of  p  others,  etc.,  the  determinant 
can  be  resolved  into  the  sum  of  vinp  •••,  etc.,  others. 

34.  Theobem.  If  the  elements  of  one  row  (or  cohnnn)  are 
equal  to  the  sums  of  the  corresponding  elements  of  the  other  rows 
(or  columns)  multiplied  by  constant  factors,  the  determinant  van- 
ishes. 

For  it  can  then  be  resolved  into  the  sum  of  a  number  oi 
determinants  which  separately  vanish.     For  example, 


mai  +  nbi    Oj 

bi 

rtj 

tti    bi 

bi     Oi     bi 

m«2  +  ^^^2      0^2 

b, 

=  m 

a.. 

a,    b. 

+  n 

&2       «■.'       ^2 

ma^  +  ?t&3    «3 

h. 

<h 

«3        ^3 

bs     «3     K 

and  each  of  the  latter  determinants  vanishes  (Art.  20). 


35.  Theorem.  A  determinant  is  unchanged  when  to  each 
element  of  any  row  or  column  are  added  those  of  several  other 
rows  or  columns,  multiplied  respectively  by  constant  factors. 

For  when  the  determinant  is  resolved  into  the  sum  of  others, 
as  in  Art.  33,  the  determinants  in  which  the  added  lines  occur 
all  vanish,  since  each  of  them  must,  when  the  constant  factor 
is  removed,  contain  two  identical  lines. 

Thus,  for  example, 


a-2    b, 
«3    h 


«i  +  mb^  +  nci 
a-j,  +  mbo  +  nc.2 
tta  +  mba  +  nc^ 


This  is  evident  since,  when  the  second  determinant  is 
expressed  as  the  sum  of  three  others,  the  two  ari.sing  from 
the  added  columns  vanish  identically  (Art.  34). 

This  proposition  will  be  found  very  useful  in  the  evaluation 
of  determinants. 


40 


THEORY  OF  EQUATIONS. 


Art.  35 


EXAMPLES. 

Find  the  value  of  the  determinant 
12      4 

2  3      7 

3  4    10 

Subtracting  the  elements  of  the  first  column  from  those  of  the 
second,  and  three  times  the  elements  of  the  first  column  from 
those  of  the  third,  we  obtain 

111 

2  11 

3  11 

which  is  identically  equal  to  zero. 


2.   Evaluate 

-111 

1 

1-1       1 

1 

1       1  -1 

1 

111 

-1 

3.   Evaluate 

7-2      0    5 

-2      6-2    2 

0-253 

= 

7-205 
19      0  -2    17 

-7      0      5-2 

f> 

19 

—  7 
12 

-2 

5 
3 

5      2      3    4 

12      0      3      9 

■972. 


Here  the  first  transformation  is  obtained  by  adding  to  the 
second  row  three  times  the  first,  subtracting  the  first  from  the 
third  row,  and  adding  the  first  to  the  fourth  row. 


Calculate  the  determinant 

1     15    14 

4 

A  = 

12      6      7 
8     10    11 

9 

5 

13      3      2 

16 

Art.  35 


PliOPEIiTIES    OF  DETEIiMINANTS. 


41 


Tlie  first  sixteen  natural  numbers  are  arranged  here  in  wluit 
is  called  a  "  magic  square,"  i.e.  the  sum  of  all  the  figures  in 
any  row  or  any  column  is  constant.  In  general,  for  a  square 
of  the  first  n^  numbers,  this  sum  is  \n(n^  +  1).  Determinants 
of  this  kind  can  be  at  once  reduced  one  degree. 

Here  adding  the  last  three  columns  to  the  first,  and  sub- 
tracting the  last  row  from  each  of  the  others,  we  have 


A  =  34: 


1  15  14     4 

16    7     9 
1  10  11     5 

=  34 

1    3     2  16 

0  12  12 

-12 

0    3    5 

rr 

0    7    9 

-11 

13    2 

16 

=  -34x12 


11-1 
3  5-7 
7  9  -11 


and  subtracting  the  second  row  from  the  last  row,  it  is  evident 
that  the  reduced  determinant  vanishes :  hence  A  =  0. 


5.   Calculate  the  determinant  formed  by  the  first  nine  natural 
numbers  arranged  in  a  magic  square: 


4  9  2 
3  5  7 
8    16 


Ans.  360. 


6.  Calculate 


2  3  8 
4  6  4 
6  12    4 


Ans.  72. 


0  111 

0  10          0 

1  0   z^  y^ 

1        0                         2^                              y2 

1  z^  0    .^•2 

1   z-    -z^     x'-z' 

1  y-  XT  0 

1  f  o?-f     -f 

1 

z- 

f 

1 

-z^ 

x»-z* 

1 

x^-f 

-.V' 

Here,  to  obtain  the  second  determinant,  we  subtract  the 
second  column  from  each  of  the  following  ones.     In  the  re- 


42 


THEORY  OF  EQUATIONS. 


Art.  35 


ducecl  determinant,  subtracting  the  first  row  from  each  of  the 
foHov/ins,  ^ve  find 


A  = 


1            z- 

f 

0        -2z- 

o(r  —  z-  —  y- 

=  - 

0     x'-.f-z'' 

-2f 

1  z-        if+z--x- 
y'  +  z^-x"  2/ 


=  {y""  +  Z--  x-f  -  4  y-z- 

=  {f  +  2^  _  .^.2  _j.  2  2/^)  (^2  +  2-2  _  ^.2  _  2  y^;) 

=  -  (a-  +  ?/  +  2)  (?/  +  2  -  a;) (2  +  x  -  y)  {x  +  y-z). 
8.    Evaluate  the  determinant 


1 
2 

L     1       4 
4     1       8 

4     12     13 

2     4     2     11 

9. 

Evaluate 

2  2      2      10 
1-1-1        5 

3  _3      3  _15 

1      1-1-5 

10. 

Evaluate 

a      b 

c 

d 

a      &         c 

d 

-a       b 

a 

/5 

0     26     c+a 

d-\-(3 

-a  -b 

c 

7 

0      0        2c 

d  +  y 

-a  -b 

—  c 

fZ 

0      0         0 

■2d 

11. 

Evaluate 

A 

= 

0  111 

1  0     a     ?> 
1     a    0     r 
1     ?>     c    0 

4» 

s.  A 

=  a-  +  6" 

+ 

c^- 

-2  he- 

A71S.   —15. 


2hibcd. 


Alt.  36 


PROPERTIES   OF  DETEUM LXANTS. 


4;{ 


MULTIPLICATION   OF    DETKU.MIXAXTS. 

36.  TiiEOKEM.  The  jrrodnct  oftico  defermiiuutts  of  any  order 
is  itself  a  determinant  of  the  same  order. 

AVe  shall  prove  this  for  two  determincants  of  the  third  order, 
and,  from  the  nature  of  the  proof,  it  will  be  evident  that  it  is 
equally  applicable  in  general. 

"We  propose  to  show  that  the  product  of  the  two  deteruiinants 

A  =  {ciih.f.-^  and  B  =  (ui/Soy-i)  is  P  = 


«!«!  +  ^1^1  4-  Ciyi     «i«2  +  ^lA  +  ("iTi    «i«3  +  ^A,  +  Ciys 
(f2«i  +  ^2/8i  +  C'2y,      (t2«2  +  ^I'iSo  +  Cryo     a.M3  +  b.J3i  +  C'sya 

«3«1  +  ^3^1  +  ^'syi        «3«2  +  ^3i82  +  ('3y2       <'3«3  +  ^Si  +  ^^m 


0) 


whose  elements  are  the  sums  of  the  products  of  the  elements 
in  any  row  of  (aiboCs)  by  the  corresponding  elements  in  any  row 
of  (fiiftoy-i)-  The  determinant  P  can  evidently  (Art.  33)  be 
expanded  into  the  sum  of  twenty-seven  others. 

The  following  proof  of  this  tlieorem  is  derived  from  Laplace's 
method  of  development  already  explained  (Art.  32). 

The  product  of  the  two  determinants.  A,  B,  is  (see  Ex.  5, 
Art.  32)  plainly  equal  to  the  determinant 


«1 

^1 

Ci 

0 

0 

0 

(k 

b. 

^"2 

0 

0 

0 

Ch 

h 

^-3 

0 

0 

0 

X 

0 

0 

«1 

(i.j 

"3 

0 

-1 

0 

/?. 

/5-. 

/?3 

0 

0 

-1 

yi 

72 

73 

(2) 


In  this  determinant  add  to  the  fourth  column  the  sum  of 
first  multiplied  by  «„  the  second  by  fi^,  and  the  third  by 
add  to  the  fifth  column  the  sum  of  the  first  multiplied  by 
the  second  by  /S.,  and  the  third  by  y.;  and  add  to  the  si 


the 
71; 

xth 


44 


THEORY  OF  EQUATIONS. 


Art.  36 


column  the  sum  of  the  first  multiplied  by  «;j,  the  second  by  ^3 
and  the  third  by  yg.     The  determinant  (2)  becomes  then 


(12  62  ^"2     02«1  +  &A  +  C2yi  a2«2  +  &2/324-C2y2  a2«3  +  &2/33  +  C273 

tts  &3  C3     a3rti  +  &3/?i  +  C3yi  «3«2+&3/?2  +  C3y2  «3«3  +  ^-^3 A  +  ^373 

-10       0  0  0  0 

0-100  0  0 

0       0-10  0  0 


And  this  is,  by  Ex.  5,  Art.  32,  equal  to  the  product,  with 
the  proper  sign  (which  in  this  case  is  evidently  — ),  of  the 
determinant 


(which  is  equal  to  —  1) 


1 

0 

0 

-1 

0 
0 

0 

0 

-1 

by  the  complementary  minor,  which  is  the  P  of  this  article. 
Hence,  the  theorem 

AxB=P. 

Cor.  Two  determinants  of  different  orders  may  be  multi- 
plied together  by  raising  the  lower  determinant  to  the  order 
of  the  higher  (Art.  30),  and  then  applying  the  above  rule. 
Thus : 


6i     c, 

xi    Vi 

bo        C2 

X 

= 

bs         Cg 

a^2    2/2 

«i 

&i 

Cl 

^2 

&2 

C2 

X 

tta 

b. 

Ca 

10     0 

0      .Ti       y, 
0     x.,     ?/2 

ai  b^x^  +  Ci?yi  b^x^  +  c{ijo 
a^  &2^i  +  C22/1  b.^2  +  C2?/2 
as     &3.r,  +  C3?/i     &3.r.,  +  G32/2 


Art.  37 


PROPERTIES   OF  DETERMINAyi'S. 


45 


37.  Eulek's  Theorem.  TJie  jyrodud  of  two  numbers,  each 
the  sum  of  four  squaj-es,  is  itself  the  sum  of  four  squares. 

By  Laplace's  method  of  development,  we  readily  prove  the 
following  identity : 


=  (a' +  b' +  c- +  d-)- 


(1) 


Similarly, 


(a'  +  l3'  +  r  +  8y   .     (2) 


a  b  c  d 

—b  a  —d  c 

— c  d  a  —b 

—d  — c  b  a 

a  13  y      S 

-(3  a  -8       y 

-y  B  a  -/3 

-8  -y  (3      a 

Now  multiply  equations  (1)  and  (2)  together,  member  for 
member. 

Letting  ««  +  b/3  +  cy-\-  d8  =  A, 

—  a/?  +  ba  -  c8  +  dy  =  B, 

—  ay  +  b8  +  ca  —  f?/?  =  C, 

—  a8-by  +  cft  +  da  =  D, 

the  product  of  the  left-hand  members  may  be  written : 


A      B      C      D 

=  {A'  +  B'+C'  +  D'f     .     (3 

-B      A  -D      C 

-C      D      A  -B 

-D  -C      B      A 

Therefore 

(a2  +  62  +  c^  +  d')  (a'  +  (3'  +  y'-\-  8")  =  (.P  +  B' +  C  +  D^, 

which  is  the  theorem.* 

*  This  theorem  is  due  to,  and  natncd  after,  the  Swiss  mathematician  I.eonhanl  Kul» 

(n07-lTS3). 

46 


TIIEOliY  OF  EQUATIONS. 


Art.  37 


EXAMPLES. 
1.    Find  the  product  of  the  two  determinants 


1    1 

1 

Xi        X2 

a?3 

Vi   y-2 

2/3 

2.    Find  the  value  of 


«ii     ai2    o, 
a,,     ««     a.. 


3.   Find  the  product  of  the  two  determinants 


1 

3    10 

3 

2    3     1 

0 

2     5     1 

' 

2 

2     2     3 

0 

1 

0 

2 

2 

0 

1 

0 

1 

0 

2 

0 

0 

9 

0 

1 

38.  Rectangular  Arrays.  Arrays  in  which  the  number  of 
rows  is  not  equal  to  the  number  of  columns  are  called  reo- 
taiif/nJar.  The  common  notation  for  rectangular  arrays,  or 
matrices,  as  they  are  called,  is : 


ttl 

&1 

tto 

b. 

} 

(f.J 

h 

b.    Co 


Eectangular  arrays  do  not  themselves  represent  any  defi- 
nite function;  but  if  two  such  arrays  of  the  same  dimensions 
are  given,  we  can  derive  from  them  by  the  multiplication 
theorem  of  Art.  36  a  determinant  whose  value  we  proceed  to 
investigate. 

(1)    When  the  number  of  columns  exceeds  the  number  of  roivs. 

Theorem.*  TJie  "2^^'oduct "  of  two  rectangular  arrays  of  the 
same  dimensions  is  equal  to  the  sum  of  the  jiroducts  of  cdl  possi- 

*  By  the  so-cullod  "jirodtR't"  hc-ro  and  the  multiplication  of  two  rectangular  arrays  in 
the  following  tlioorcm,  we  simply  mean  that  the  procfxn  of  Art.  36  is  employed  ;  of  course, 
as  matrices  are  not  functions,  they  cannot  really  be  multiplied  together. 


Art.  38 


PBOPERTIES   OF  DETKIiMlNAyTS. 


47 


ble  deter)ni)iaHts  ichich  can  be  formed  fruni  one  army  [In/  tnkimf 
a  number  of  columns  equal  to  the  number  of  j-ows)  multijilied  by 
the  correspondimj  determinants  formed  from  the  other  array. 


To  prove  this,  take  any  two  rectangular  arrays, 

«!     &i     t'l     (?i         ...  «i     A     yi     ^1 

ttj     62     *^-2     (h  '  «■■•     ^2     72     82 


(2) 


and  perform  on  these  a  process  simihir  to  that  employed  in 
multiplying  two  determinants.  We  thus  obtain  the  deter- 
minant 

o.«i  +  60^1  +  c.yi  +  fUi       UM..  4-  ^2^3.  +  f.y^  +  dA, 
The  vahie  of  this  is  easily  found  to  be 

(«.^2)  («iA)  +  («iC2)  («iy2)  +  ((hd-d  («iS2)  +  (^c•2)  OS.y,,) 

Hence  the  theorem.     This  proof  can  be  easily  generalized. 

(2)  When  the  number  of  roivs  exceeds  the  number  of  columns. 

Theorem.     In  this  case,  the  determinant  resultinrf  from  the 
multiplication  (so  called)  of  the  two  arrays  vanishes. 

Take,  for  example,  the  two  arrays, 


«1 

hi 

a^ 

b. 

as 

h 

(1), 


«1 

/?. 

«2 

^2 

(^ 

^S 

(2). 


Performing   the   process   of    multipli(!ation,    we    have    the 
determinant 

«,«1  +  b^/Si       «i«2  +  ^1)^2       «1«3  +  Wf^s 

a^fCi  +  b.,(3i     a.2(i-2  +  ^Si     cue,  +  b.fi^ 

a3«l  +  h(^\       «.i«2  +  ^3/^2       ".!'<3  +  Ms 


48 


THEORY  OF  EQUATIONS. 


Art.  38 


This  determinant  is  obviously  the  same  as  would  arise  if  a 
column  of  ciphers  were  added  to  each  of  the  given  arrays,  and 
the  determinants  so  formed  then  multiplied.  It  follows  that 
the  determinant  vanishes. 

In  an  exactly  similar  way,  we  can  prove  the  general  theorem. 

I  ^  EXAMPLES. 

1.    From  the  two  arrays 


111' 
a    (3    y 


(1), 


111 

«     /3    7 


(2), 


prove 

3  a-\-(3  +  y 

a  +  (3-hy     a'  +  l3-  +  y- 

2.    By  squaring  the  array 


^(a-(3y-  +  ia-yy  +  (f3-yy 


a     b     c 

d     e    f 
prove 

{a-  +  b--Y-c')  (d-+e^+f-)  =  (ad  +  &e  +  c/)-+(ae  -  bdy-\-(cd  -  a/y 

+  (bf-cey. 

39.  Reciprocal  Determinants.  The  first  minors  (with  their 
proper  signs)  xl„  Bj,  Ci,  •••  A2,  B.^,  etc.  (Art.  22),  which  occur 
in  the  ex})ansion  of  a  determinant  are  called  inverse  elements; 
and  the  determinant  formed  with  them  as  elements  is  called 
the  inverse  or  reciproccd  of  the  original  determinant.  The 
following  theorem  gives  a  useful  relation  connecting  the  two 
determinants : 

Theorem.  The  reciprocal  of  any  determinant  of  the  ni\\  order 
is  equal  to  the  {n  —  l)th  power  of  the  given  determinant. 

Let  the  reciprocal  of  A  be  denoted  by  A',  and  multiply  the 
two  determinants 


Art. 


PROPERTIES   OF  DETERMIXAXTS. 


4!» 


A  = 


«1 

h 

C-i 

a. 

b. 

Co 

a. 

h 

Cs 

A': 


A    ^1    c, 

-I2       ^2        C'i 

A,    B,     6-3 


All  the  elements  of  the  resulting  determinant  except  those 
in  the  diagonal  vanish  (Art.  29) ;  and  the  result  is 


AA' 


AGO 
0  A  0 
0     ()     A 


whence 


From  the  nature  of  the  above  proof,  it  is  evident  that  the 
process  here  employed  in  a  particular  case  is  equally  appli- 
cable in  general ;  giving  for  a  determinant  of  the  nt\i  order 

AA'  =  A'',    or    A'  =  A"-^ 


EXAMPLES. 
1.    If  A'  =  the  reciprocal  of  the  determinant 


12    3 

-11 

9 

6 

3    14 

,  show  that  A'  = 

2 

-13 

8 

6    4    5 

5 

5 

-5 

and,  hence,  verify  the  fornmla  A'  =  A-. 
2.    Form  the  reciprocal  of  the  determinant 

A  = 


a 

h 

9 

h 

h 

f 

9 

f 

c 

CHAPTER   III. 


APPLICATIONS   AND    SPECIAL    FORMS   OF    DETER- 
MINANTS. 

APPLICATIONS  OF   DETERMINANTS. 

In  Arts.  9,  11,  12,  and  14,  we  have  seen  how  the  work  of 
solving  simple  linear  equations  of  two  or  three  variables  may 
be  abbreviated  by  the  use  of  the  determinant  notation.  We 
shall  now  extend  these  principles,  and  proceed  to  investigate 
some  of  the  fundamental  properties  of  systems  of  equations. 

40.  First,  taking  a  special  case,  let  it  be  required  to  solve 
the  simultaneous  linear  equations, 


cii'x'  +  ai"x"  +  ai"'x"'  =  u^ 
a.,'x'  +  02";c"  +  a.,"'x"'  —  n.^ 
tts'x'  +  0,3  "ar"  +  a3"'x"'  =  u^ 


Cli' 

Cli" 

a, 

a.J 

a.," 

«2 

a' 

a," 

a. 

(1) 


(2) 


is  called  the  determinant  of  this  system  of  equations. 
By  Art.  29,  we  have : 

A.'ni'  +  ^aV  +  .43'a.3'  =  A, 

A.'cii"  +  AJaJ'  +  AJcis"  =  0, (3) 

A,'a,"'  +  AJaJ"  +  A,'a,"'  =  0, 

If  now  we  add  the  equations  (1)  after  having  multiplied 
them  respectively  by  J/,  A^,  A^',  the  coefficient  of  x'  would 
60 


Alt.  41 


become  A,  and  those  of  x"  and 
have 


iPPLICATIOyS   OF  DKTEUMIXAXTS.  51 

become  zero.     Hence  we 
Ax'  =  Ai'ui  +  A,'u.,  +  A^'u^ 


«1 

0," 

0/" 

»2 

«.," 

a,'" 

"3 

%" 

a-J" 

»1 

«l" 

or 

»2 

a.," 

a,'" 

«3 

«3" 

a,'" 

a/ 

0/' 

ar 

0,' 

CI2" 

a.,'" 

«3' 

«3" 

a.J" 

The  values  of  x"  and  x'"  may  be  found  in  the  same  manner. 

We  proceed  in  exactly  the  same  way  to  solve  the  general 
case,  as  follows. 

41.   Let  the  given  system  of  simultaneous  linear  equations  be 
a^x'  +  0/ V  -\ \-  o/'V"  H f-  ai'">a;(">  =  «, 


a.^x'  +  a^'x"  +  •••  +  «2<''*.f"'  H h  o,.*"'.r<"' 


(1) 


rt„'.x-'  +  a„"a;"  -\ 1-  a,/".i-<"  H \-  a„<"'.i-<"'  =  »„ 

where  the  number  of  unknown  quantities  is  the  same  as  the 
number  of  equations.  Let  us  form  the  determinant  of  this 
system  of  equations 


A  = 


ai      a, 
a  J     a., 


(-') 


«,.         '^H       •"  ''..       •••  "n 

and   let  A,^'^   be  the  coetficicnt   of   a^'"    in  this  doterminant. 


52 

The  sum 


THEORY   OF  EQUATIONS. 


+  A,^'W^>  +  -  +  A/W 


Art.  41 


(3) 


is  equal  to  A  for  j  =  i,  and  is  zero  for  all  values  of  j  different 
from  i.     (Compare  Art.  29,  Cor.) 

If  now  we  add  the  equations  (1),  after  having  multiplied 
them  respectively  by 

A,^'\AJ'\--AJ'\ 

the  coefficient  of  a;'"'  is  equal  to  A,  and  those  of  all  the  other 
unknown  quantities  vanish.     We  have  therefore 

•     Ao;"'  =  ^i"'«i  +  A^^'hio  -\ 1-  A"*"n 


Ul        «! 


«/ 


(i) 


the  second  member  being  what  A  becomes,  when  we  replace 
the  coefficients 

of   x''*   by  the   second   members  of   the  corresj)ondiug  given 
eqiiations 

As  long  as  A  is  different  from  zero,  this  formula  gives  for  the 
n  unknown  quantities  finite  and  determinate  values. 

Cor.  If,  for  brevity,  we  denote  the  numerator  of  the  frac- 
tion giving  the  value  of  x''^  (Equation  4)  by  8*",  then,  with  this 
notation,  we  would  have : 


8'       „      8" 

— ,  x"  =  — 

a'  a 


8(n) 


Now,  if  A  =  0,  and  8',  8",  •••  8'"'  are  not  zero,  then  the  values 
of  the  unknowns  are  infinite. 


Art.  41 


APPLICATIONS   OF  DETEIlMiyANTS. 


53 


If  A  =  0,  and  at  the  same  time  S'  =  0,  8"  =  0,  etc.,  tlicn  tlie 
values  of  the  unknowns  are  indeterminate.  This  would  be 
the  case  if  Ui  =  0,  ?<2  =  0,  •••  u„  —  0. 


EXAM  PLES. 
1.    Solve  the  equations 

x  +  rj  +  z  +  t+tc=  5 

x+y+z+t+v=  3 

x+y+z+u+v=  1 

x  +  y  +  t-}-u-i-v=  7 

x  +  z  +  t  +  ic  +  v=  9 

y-{-z  +  t+u  +  v  =  11 

Here  there  are  six  equations  and  six  unknowns,  and  as  A  is 
not  zero,  as  we  find  by  calculation,  there  is  a  solution.  We 
first  calculate  A,  and  then  the  determinants  which  we  may  call 

8.,  8„  8,,  8,.  S,„  8,   ^  ^    \AK,S^    y      '        -   r: 

111110  ^ 

11110  1 

1  1  1  0  1  1 

110  111 

10  1111 

0  1  111^ 


^N 


0 

0 

0 

0 

1 

-1 

0 

0 

0 

1 

-1 

0 

0 

0 

1 

-1 

0 

0 

0 

1 

-1 

0 

0 

0 

'  1 

-1 

0 

0 

0 

0 

0 

1 

1 

1 

1 

1 

0 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

-1 

0 

0 

0 

0 

5 

5 

4 

3 

o 

1 

+  0 


54 


THEORY   OF  EQUATIONS. 


Art.  41 


In  reducing  this  determinant,  we  have  employed  the  princi- 
ple of  Art.  35.  As  a  still  further  illustration  of  the  ready- 
application  of  this  principle,  we  give  the  steps  in  the  calcu- 
lation of  8,. 


K  = 


1 
1 

1 

0 

0 

0 

1 

1 
1 

= 

1 

1 

1 

1 

5  1111 

2      0      0     1-1 

6  0-11      0 
8-1      0    1      0 

10      0      0     1      0 


13      1 

8  -1 
10      0 


5  11110 
2  0  0  1-11 
0      0      0    0      0    1 

6  0-11  0  1 
8-1      0     1      0    1 

10      0      0     1      0    1 

5  1111 
7      112    0 

6  0-110 
8-1      0    10 

10      0      0    10 


7  1 
13      1 

8  -1 


10      0    0 


13  1  3 
21  0  4 
10    0    1 


=  -19. 


15  1 
13  1 
111 
17  0 
19  1 
0  11     1 


0  1 

1  1 


=  -9 


■^ 


^.1 


o 


Art.  42        APPLICATION.'S   OF  UETEIiMiyANTS.  55 

Similai'ly 

8,  =  +  l,     8,  =  +  31,     8„=  +  21,     8„  =  +  ll 
Therefore  we  have 

2.  Solve  the  system  of  equatiuus, 

—  Xi  +  a;,,  +  Xs  +  x-4  =  8, 
a'l  —  .To  +  .Tg  +  x^  =  6, 
cc,  +  X,  —  a-g  +  Xi  =  4, 
Xi  +  x.^  +  x^  -  a:^  =  2. 

3.  Solve  the  simultaneous  equations 

x-2y-{-Zz=  6, 
2x  +  3y-4z  =  20, 
3x-2y  +  5z  =  26. 

A)is.   x  =  8,  ,v  =  4,  2  =  2. 

42.  Number  of  Equations  Greater  than  the  Number  of  Un- 
knowns. In  this  case  where  the  number  of  equations  in  a 
given  system  is  greater  than  the  number  of  unknowns,  it  will 
not,  in  general,  be  possible  to  solve  the  system.  Whenever 
values  may  be  assigned  to  the  unknowns  which  \vill  simul- 
taneously satisfy  all  the  equations,  the  system  is  said  to  be 
consistent.  The  consistency  of  any  such  system  must  obviously 
depend  upon  some  relation  among  the  coefficients. 

We  shall  first  find  what  this  relation  is  for  the  simple  case 
where  we  have  three  simultaneous  equations  involving  only 
two  unknowns. 

Let  the  given  equations  be 

aix'  +  bix"  =  k^ (1) 

a^' +  b^x"  =  k. (2) 

a3x'  +  b,x"  =  k, (3) 


56 


THEORY  OF  EQUATIONS. 


Art.  42 


Since  the  above  system  is  to  be  consistent,  the  values  of  the 
unknowns  obtained  by  solving  any  two  of  the  equations  must 
satisfy  the  third  equation. 

Solving  equations  (^2)  and  (3),  we  get 


h. 

b. 

b. 

k. 

1  _ 

h 

h 

=  - 

A^ 

h 

a.2 

b. 

Ch 

~h 

«3 

hs 

«3 

h 

a.2 

k2 

«3 

h 

a.2 

h 

«3 

h 

S'ubstituting  these  values  of  x'  and  x"  in  equation  (1),  and 
reducing,  we  get 


«i 


b.2 

h 

-b 

1  1  ^2   ^'2 

+  k. 

ao 

b. 

h 

ks 

1    , 

1  «3  ks 

as 

6 

«i  bi     ki 

«2   &2   ^'2 

=  0, 

«3   ^3   ^ 

3 

0, 


which  is  the  condition  of  consistency  of  the  three  given  equa- 
tions.    For  example,  the  system  of  equations 

6a;'+  x"  =  —  l, 
5.x' -10 a;"  =  5, 
4.r'+    3a;"  =-7 

is  consistent,  because  we  have 


=  0. 


43.  We  shall  now  take  up  the  general  case,  and  investigate 
this  relation  in  the  case  of  {n  +  1)  linear  equations  involving 
n  unknowns. 

Consider  the  following  system  : 


Art.  43  APPLICATION.'^   OF  DETERMINANTS. 


57 


a^x'     +  a,  .r 
a.,'x'     +  a.,"x' 


+  ••• +  a,"'>x<">     =«, 


a„'.f'     +  ((J'x"     +  •••  +  a,/"'.f*"'     =  u„ 


a) 


Since  the  above  system  is  to  be  regarded  as  consistent,  the 
values  of  the  iinknowns  obtained  by  solving  any  ?t  of  the 
equations  must  satisfy  the  remaining  equation. 

Solving  the  last  n  equations  by  the  method  of  Art.  41,  we 
obtain,  after  permuting  the  column,  ?<,,  »3  •••«„+!,  till  it  occu- 
pies the  last  position,  and  having  regard  to  the  proper  signs : 


x'=i-ir-^ 


x"=(-iy- 


.(n) 

a. 
a 

1 
1+1 

«2 
'   «„+! 

a.i' 

a,"      -a,*'" 

n) 

««+ 

,'     «,.+i"  •••««+■ 

58 


THEORY  OF  EQUATIONS. 


Art.  4.3 


Substituting  these  values  in  the  first  equation  of  system  (1), 
clearing  of  fractions,  and  reducing,  we  obtain 


a/ 

•••     a/"' 

«i 

tta' 

...     «,'»> 

''2 

aj 

...     «„'"> 

u„ 

a„+i' 

"•     «„+/" 

w„+l 

=  0 


(2) 


which  is  the  condition  of  consistency,  giving  the  required  rela- 
tion among  the  coefficients. 

AVhen  the  equations  are  consistent,  this  determinant  is  called 
the  eliminont  or  resnltant  of  the  system,  because  it  is  the 
result  obtained  by  eliminating  the  unknowns  from  the  given 
equations. 

We  should  observe  that  the  resultant,  in  this  case,  is  the 
determinant  of  the  coefficients  and  absolute  terms. 

Example.     Test  the  consistency  of  the  system 

a; +  15?/ +  14  2=    4, 
x+    6t/+    lz=    9, 


Here 


X 

+  102/ +  11  2!  = 

5, 

x-\-   3y+   22  =  16. 

1     15    14      4 

16      7      9 

" 

1     10    11      5 

1      3      2    16 

and  the  system  is  consistent. 


HOMOGENEOUS   LINEAR   EQUATIONS. 

44.  If  in  equations  (1)  of  the  preceding  article,  the  absolute 
terms  (it's)  become  zeros,  we  have  a  system  of  homogeneous 
linear  equations,  and,  in  this  case,  the  numerators  of  the  frac- 


Art.  44         HOMOGENEOUS   LINEAR   EQUATIONS. 


r)9 


tions  giving  the  values  of  llie  unknown  quantities  vanisli. 
This  shows,  as  we  know  from  other  considerations,  that  such  a 
homogeneous  system  can  always  be  satisfied  by  giving  to  each 
unknown  the  value  zero.  It  often  happens,  however,  that  such 
equations  may  be  simultaneously  satisfied  by  assigning  to  the 
unknowns  values  other  than  zero. 

We  shall  now  consider  the  case  of  a  system  of  n  homo- 
geneous linear  equations  involving  n  unknowns. 


Let 


ai'x'  +  cii'x"  +  •••  -f  o/ 


=  0 


a^'x'  +  a.J'x"  +  •••  +  o,,("'.^<"'  =  0 
a„'x'  -f  a„"x"  H h  a,/"'.r'"'  =  0 


•     •     (1) 


be  any  system  of  n  homogeneous  linear  equations  involving  n 
unknowns  x',  x",  -■'  x^"\  in  which  the  coefficients  are  so  related 
that 

a  I     cii"     ■"    cf/ 

aJ     a.>"     •••    a.7* 


a,'    a„ 


a,. 


0 


(2) 


Applying  the  method  of  Art.  41  to  the  system  (1),  we  can 
obtain  the  values  of  the  unknowns  only  in  the  indeterminate 
form  -.     (Compare  Art.  41,  Cor.) 

Though  it  is  thus  impossible  to  determine  the  absolute 
values  of  the  unknowns  in  such  a  system  as  (1),  it  is  possible 
to  find  the  ratios  of  any  (n  -  1)  of  the  unknowns  to  the 
remaining  one. 

For,  dividing  each  of  the  equations  (1)  by  a;<",  and  repre- 
senting the  ratios 


x^*^    x"* 


^,  ...^_  by  v',v", 


60 


THEORY  OF  EQUATIONS. 


Art.  44 


respectively,  rememberiug  that  i'^''  =  1,  we  obtain  the  system 


(3) 


'y" +  •••+«„ 


'+a„ 


'  +  •••  +  «/ 


'=-a('> 


This  is  a  system  of  n  non-homogeneous  linear  equations 
involving  (?i  —  1)  unknowns,  v',  v",  •••  v'-'~'^^,  v^''^^'',  •••  t'^"^,  and, 
so  long  as  the  condition  of  consistency  (2)  holds  good,  as  in 


■1),  v^'^ 


may 


Art.  43,  the  values  of  the  ratios  v',  v", 
generally  be  obtained  by  solving  any  n  —  1  of  equations  (3). 
Hence  the  system  (1)  will  be  satisfied  by  any  values  of  x',  x", 
•  ••  x^"^  among  which  we  have  the  ratios  v',  v",  •••  v''"\  as  deter- 
mined by  any  n  —  1  of  equations  (3)  ;  that  is,  if  equati8fei-(l) 
are  satisfied  by  the  values  Xq',  Xq",  •••  a;o^"',  they  will  be  equally 
satisfied  by  Xxq,  Xxq",  •••  AaV"',  A  being  any  factor. 

The  relation  (2)  gives  the  condition  of  consistency  of  the 
system  (1),  and  A,  tJie  determinant  of  the  coefficients,  is  the  elimi- 
minant  or  resultant  of  the  system. 

To  illustrate,  let  us  solve  the  homogeneous  system 

2  a; +  4^  +  52  =  01 
3x-]-5y  +  6z  =  0  \ 
4,x  -{-  6y  +  7 z  =  0  ) 
Here  the  determinant  of  the  coefficients 
2    4    5 
A=    3    5    6    =0. 
4    6     7 
An  attempt  to  solve  the  system  (1)  by  Art.  41  gives 

0 
0' 

which  are  indeterminate.     But,  since  A  =  0,  the  system  is  con- 
sistent, and  we  can  obtain  definite  values  for  the  ratios  ?,    ^- 


(1) 


0 
0' 


0 

z  =  -, 
0 


I  0 


Art.  45 


IIOMOGEXEOUS  LINEAR    EQUATIoys. 


01 


Dividing  the  equations  by  z,  we  have 


2-  +  4'l  =  -5 

z        z 


3^  +  5-^  =  -6 

z        z 


4^  +  6-^: 

z        z 


(2) 


Solving  any  two  of  these,  we  get 


m 


z~     2'    'z~     2'    "^  '    ll 
x:}j:z::l:-3:2, 
tities  having  these  ratios  will  satisfy  the  given 


ecjiiations. 


45.  By  the  last  article  the  system  of  homogeneous  linear 
equations 

a^'x'  +  ai"x"  +  •••  +  a/"\t;("'  =  0 
a.,'x'  +  aj'x"  +  •••  +  02<"'.«<"'  =  0 


.     .     (1) 


ajx'  +  a„"x"  +  •••  +  a,.<"\i;'">  =  0 

is  consistent  if  A  =  0,  and  in  this  case  we  can  determine  the 
values  of  the  (n  —  1)  ratios 


x^  '     x^  '        x^  ' 
Now,  since  A  =  0,  we  have  (Art.  29)  the  n  equations 
a/A'  +  (h"A"  +  "•  +  ",'"'^1*""  =  0, 

a;  A'  +  a,"  A,"  +  -  +  a*'"'.-l*<"'  =  A  =  0, 

a,:  Ax'  +  a„"J^"  +  •••  +  a,;"».l<^""  =  0, 


62  TEEOUY  OF  EQUATIONS.  .  Art.  45 

which  give  for  the  ratios 

values  identical  Avith  those  which  the  proposed  equations  (1) 
give  for  the  ratios 

x[_     x"       -r'"  ''. 

therefore  A^',  A^"  •••  ^4'"'  are  proportional  to  x',  x"  •■•  a;*"'  what- 
ever may  be  the  index  Jc,  so  that  we  have  the  proportions 

x':x":  ...:a;(">  =  A':^":  " 
=  Ao':A'':- 


=  A,::AJ':  - 

Hence,  in  any  determinant  lohich  equals  zero,  the  mino^of  the 
elements  in  any  row  (or  column)  are  jwoportional  to  the^kivors 
of  the  corresponding  elements  in  any  other  row  (or  column). 

46.  Among  the  proportions  of  Art.  45,  let  us  consider  those 
of  the  last  line,  for  example 

x':x":-:  .7j(»)  =  A„' :  A,"  :  -  :  AJ'^K      ...     (1) 

The  coefficients  of  the  last  of  the  proposed  equations  ((1)  of 
Art.  45) 

a„'x'  4-  a„"x"  +  •••  +  a,/"'.c("'  =  0, 

not  appearing  in  the  expressions  for  A  J,  AJ',  •••,AJ"\  there 
results  that  the  proportions  (1)  determine  the  ratios-of  the, 
unkiiowns  a;',  x",  •••,  x'"\  which  will  satisfy  the  n  —  1  equations 

«i'.i;'  +  ai"x"  -\ h  a/"'a;(")  =  0 


Alt.  n 


DKTEIiMINANTS   OF  SPECIAL    FOI{^fS. 


63 


expressed  by  means  of  the  niiiiovs  wliich  can  be  formed  with 
the  »(h  — 1)  coefficients  of  these  equations,  in  suppressing  in 
turn  each  of  the  vertical  lines.  Therefore  having  given  n 
homogeneous  equations  between  u  +  1  unknowns 


ajx'  +  aj'x"  H 1-  a„<"+"x"'+"  =  0 

,'  •••o/'-^'     a/'^"  ••.o,<" 


if  we  put 


a '■" 


«« 


the  solution  of  the  proposed  equations  would  be  given  by  the 
proportions 

x'  :x":--:  .t'"+"  =  B' :  R"  :  ■■•  :  /2<"+»>. 


For  example,  the  two  equations 

-4:x  +  y  +  z  =  0 
x-2y+z=0 


give 


x:y:z  = 


1  1 

2  1 
=  3:5:7. 


-4     1 
1     1 


4      1 
1  -2 


0   y  DETEinilXAXTS   OF   SPECIAL   FORMS. 

47.  Symmetrical  Determinants.  Two  elements  of  a  deter- 
minant so  situated,  that  one  occupies  with  reference  to  the 
leading  element  the  same  position  in  the  rows  as  the  other 
does  in  the  columns,  are  called  ro>>jin/((te  elements.  For  ex- 
ample, in  the  common  form  of  determinant,  d^  and  /v^  are  con- 
jugates, one  occui)ying  the  fourth  i)lace  in  the  second  row,  and 
the  other  the  fourth  place  in  the  second  column. 


64 


THEORY  OF  EQUATIONS. 


Art.  47 


Each  of  the  leading  elements  (that  is,  the  elements  of  the 
X)rincipal  diagonal)  is  its  own  conjugate.  Any  two  conjugate 
elements  are  situated  in  a  line  perpendicular  to  the  principal 
diagonal,  and  at  equal  distances  from  it  on  opposite  sides. 

A  symmetrical  determinant  is  one  in  which  each  element 
has  itself  for  a  conjugate  element.  Examples  of  symmetrical 
determinants  are  the  following : 


a     h     g 

h     b    f 

g  f   c 

(1). 

0 


y   X 


(2) 


In  a  symmetrical  determinant  the  first  minors  complementary 
to  any  two  conjugate  elements  are  equal,  since  they  differ  only 
by  an  interchange  of  rows  and  columns.  The  corresponding 
inverse  elements  are  also  equal,  the  signs  to  be  attached  to  the 
minors  being  the  same  in  both  cases.  It  follows  that  the  recip- 
rocal of  a  symmetrical  determinant  is  itself  synimetrical. 

The  leading  minors  are  all  symmetrical  determinants. 

The  principal  diagonal  is  called  the  axis  of  symmetry. 


EXAMPLES. 

1.  Form  the  reciprocal  of  the  symmetrical  determinant 
a    h    g 
h     b    f 
9    f    c 

Using  the  capital  letters  to  denote  the  reciprocal  elements 
(Art.  39),  the  reciprocal  determinant  may  be  written  thus : 

A     H    G  hc-f-     fg-ch     hf~bg 

A'  =     H    B     F     =    fg  -  rh     ca  -  f/     {/''  -  «/ 

G    F     C  hf-bg     gh-af    ab-h? 

2.  Prove  by  means  of  the  proposition  of  Art.  36,  that  the 
square  of  any  determinant  is  a  symmetrical  determinant. 


Art.  18         DETERMINANTS   OF  SPECIAL    Foil  MS. 


r,5 


48.  Skew-Symmetric  and  Skew  Determinants.  A  skeir-st/m- 
metric  determinant  is  one  in  which  each  element  is  its  conju- 
gate with  sign  changed.  Since  each  leading  element  is  its 
own  conjugate,  it  follows  that  in  such  a  determinant  all  the 
elements  of  the  principal  diagonal  are  zero.  For  example,  the 
determinant 


A  = 


0 

a 

h 

a 

0 

d 

b 

-d 

0 

-c  -e  -f 


is  skew-symmetric. 

A  slcew  determinant  is  one  in  which  each  element,  except  the 
leading  elements,  is  its  conjugate  with  sign  changed. 

Thus,  while  a  skew-symmetric  determinant  is  zero-axial,  a 
skew  determinant  is  not.     Thus 


b 

c 

I 

'nl 

z 

n 

n 

10 

is  a  skew  determinant. 

MISCELLANEOUS   EXAMPLES. 
Evaluate  the  following  determinants  : 


2     1     10 

2. 

13    0 

3. 

25     5     10 

3    0      6 

2    0    5 

15    3      9 

4    5      7 

4    6    7 

12      3 

10    4    5 

5. 

10    0    5 

6. 

1     3    4 

15    5    6 

15    3    6 

2     4     5 

20     6    8 

20    4    7 

3    5    6 

66 


THEORY  OF  EQUATIONS. 


Art.  48 


10. 


13. 


3  4        5     I 

4  -1  -2 
0  3  0 
3-7      4 

15  13     10 
12     17     10 

16  11     19 


11. 


14. 


4 

5 

2 

-1 

2 

-3 

6 

-4 

5 

-1 

-1 

1 

-3 

1 

-4 

2 

-3 

-5 

20 

15 

25 

17 

12 

22 

19 

20 

16 

12. 


15. 


1  - 

-1 

1 

4  - 

-3 

0 

3 

9  _ 

5 

15 

17 

16 

12 

18 

14 

19 

17 

13 

30 

36 

35 

33 

31 

37 

38 

34 

32 

16.   Expand  and  simplify  the  determinant 
a        a  +  3     a  +  6 

a  +  1     «  +  4     a  +  7 
a +  2     a  +  5     a +  8 

Evalnate  the  following  determinants : 
17. 


Ans.  0. 


19. 


21. 


Ill: 

L 

18. 

2 

3 

-1      . 

5 

1-1-1     1 

0 

6 

-5  -3 

1-1     1-1 

1 

1 

1      1 

1     1-1-1 

Ans 

+  16. 

1 

-1 

1  -1 

An& 

.  -74. 

2      2      2    10 

20. 

6 

3 

2     1 

0      0      12 

5 

8 

7     2 

3      4-32 

4 

o 

8     4 

1-14      5 

Ans 

.  +16. 

3 

6 

3    3 

A7 

IS.   +660 

1     3     5     2 

22. 

0 

0 

0    4 

12     5     1 

0 

0 

1    5 

2    4    4    3 

0 

3 

2    4 

5    2     2    1 

6 

2 

0    3 

Art.  48 

3/ 

23. 

3 

1 

4 

2 

2 

8 

1 

6 

4 

3 

9 

5 

MISCELLANEO  US    EX  A  MPL  ES. 


25. 


27. 


29. 


31. 


3  4 

■3  1 

6  -2 

5  9 

1  -3 


24. 


4hs.  -101. 


2 

1 

1 

2 

0 

3 

4 

0 

5 

2 

2 

5 

0 

0 

7 

5 

2 
3 
4 
1 
4 
Ans.  172. 


1  1 
1  0 
0 


5 
3 

1 
7 

0  9 

1  11 


^)iS.  ^ 


26. 


30. 


32. 


3    7    4 

3 

7    4    3 

5 

2    19 

4 

8    G    4 

7 

vl>/.s.  -33(5. 

10      8 

9     14  1 

17     15 

18     11 

15    19 

10     13 

16    17 

18     10 

Ans.  -2660. 

5  -1 

4      6-2 

-1      4 

6  -2       5 

4      6 

-2      5  -1 

6  -2 

5-14 

-2      5 

-1       4      6 

Ans.  +22,692. 

0      3 

7         9    5 

-3      0 

4          2     1 

-7  -4 

0        11  101 

-9-2 

-11       0    2 

-5  -1  - 

-101  -2    0 

2     4 

3     14     3 

-4     2  - 

-3     2-1     2 

5-1 

6     2-1     5 

1      1 

1  —2  —2  —2 

7  -3  - 

-5      1      4      2 

3     1 

2  _ 

1 

2     3 

^ns.  +14,940. 


68 


THEORY   OF  EQUATIONS. 


Ai-t.  48 


12 

22 

14 

17 

20 

10 

16 

-4 

7 

1 

—2 

15 

10 

-3 

o 

3 

—2 

8 

7 

12 

8 

9 

11 

C 

11 

2 

4 

-8 

1 

9 

24 

6 

6 

3 

4 

22 

33. 


.^jis.  12,228. 

34.  Find  the  number  of  inversions  in  the  series 

h  a  c  f  i  g  d  h  e. 

35.  Find  the  number  of  inversions  in  the  foHowing  permu- 
tations : 

3,  6,  4,  1,  5,  2 ; 

7,  1,  6,  5,  3,  4,  2 ; 

2,  4,  1,  3,  6,  7,  5 ; 

4,  8,  6,  7,  2,  5,  3 ; 

3,  1,  8,  9,  2,  5,  6,  7,  4. 

Develop  the  following  determinants : 


36. 


38. 


X 

0 

y 

0 

X 

y 

- 

X  - 

-y 

0 

0 

d 

d 

d 

a 

0 

a 

a 

b 

h 

0 

b 

c 

c 

c 

0 

37. 


39. 


a 
-1 

0 
0 


0 

0 

1 

0 

c 

1 

1 

d 

Ans.  abed  -{-  ab  +  orf  +  cd  +  1. 


40. 


1+a  111 
1  1+6  1  1 
1  1      1+c      1 

111     1+d 
Ans.  abcd(l  -\-  a~^ 


+  b-'  +  C-'  +  d-'). 


'^'' 


Art.  48 

MI  SC ELLA  NEO  US 

EXAMl'LES. 

69 

41. 

0     0     k     I     X 
0    0    h    X   0 
0    0    .T    0    0 
0     X    e    f   g 

42. 

tti    a.,    a^    a^ 
0     6,   0     0 
0     c,    C-,    c, 
0     f/o    0     iU 

X    a     b     c    cl 

Ans.  x' 

Alts.  a^b■Jr./l^. 

43.   Write  the  expauded  form 

of  the  determinants : 

1  a-^aj'tt^' 

1; 

1  an022«K« 

vif^oo  1 ; 

2  ±  ciyb.:^ 

Yh^s- 

Find  the  values  of  x  in  the  following 

equations : 

44. 

X  -i       1 

45. 

1     1     1 

-6       3  -2 

=  0. 

a     X     c 

=  0. 

a;      2       1 

b     b     X 

46. 

a  +  bx    c    d 

47. 

0      X      3 

e  +fx    g    h 

=  0. 

1  -X      4=0. 

i  +  kx   I    in 

2      5-6 

Ans.  x  = 

1  f<^»*'  1 
1  bgm  1 

^ 

L>(S. 

x  =  -l 

48.   "What  effect  is  produced  on  a  determinant  of  the  jjth 
degree  by  multiplying  all  its  elements  by  —  1  ? 


49. 


50. 


Show  that 

1 

1 

1        1 

1 

l+o; 

1        1 

1 

1 

1+y       1 

1 

1 

1            1+2 

Show  that 

0      a.^ 

«3 

oil 

bi     &2 

b,       = 

1     f/;,///-,     aM,c, 

Cl 

C2 

Cs 

1     tta^^iCj     a. 

'V3 

70 


TUEORY  OF  EQUATIONS. 


Art.  48 


51.   Prove  that 

. 

a  +  c 

b  +  d 

a  +  c 

b  +  d 

b+d 

a  +  c 

b+d 

a+  c 

a  +  b 

b  +  c 

c  +d 

d+a 

c  -\-d 

d  +  a 

a  +  b 

b  +  c 

52.    Showth 

It 

0 

a 

b 

c 

a 

0 

c 

b 

b 

c 

0 

a 

" 

c 

b 

a 

0 

0 

1 

1 

1 

1 

0 

c' 

6^ 

1 

c' 

0 

a^ 

1 

b' 

a' 

0 

=  0. 


Resolve  into  simple  factors  the  two  determinants ; 


63. 


a;  1  1  1 
1  CB  1  1 
11x1 
1     1     1     x 

Ans.  {x+3){x-iy 


54. 


a  a  a  a 
a  b  b  b 
a  b  c  c 
a     b     c     d 

ins.   —a{a—b)(b—c){c—d). 


a 

b 

c 

d 

b 

c 

d 

a 

c 

d 

a 

b 

d 

a 

b 

c 

55.   Transform 


so  as  to  have  the  principal  diagonal  composed  (1)  of  the  four 
a's,  (2)  of  the  four  b's,  (3)  of  the  four  c's,  (4)  of  the  four  d's. 

Prove  the  following  identities : 


56. 


a  +  b 
a 
b 


c  c 

b  +  c        a 

b        c  +  a 


Aabc. 


Art.  48 


MISCELLA NEO US   EXAMPLES. 


71 


57. 


58. 


c 
a 


c  c 

b'  +  '•- 


a 
b  b 

a  -^b  —  c  c 

a  b  +  c 

b  b 


a 

C-  +  a- 


59.   Find  the  value  of 

12  11 

3    0  14 

0    2  1-1 

2    3  0-4 


4  abc. 


c 

a 

b 

c 

a 

a 

= 

b 

c 

a 

c^  a  —  b 

c 

a 

b 

-1 

9 


4  2-1 
_  1  3  -2 
0  2-1  1 
3      0      4-1 


Perform  the  following  multiplicatious,  giving  the  results  as 


determinants : 


60. 


61. 


b    0 

0    d 

e    f 

Ans. 


a 

b 

0 

d 

/ 

0 

ab 

ac 

ae  +  bf 

Ans. 

bd 

c'  +  d' 

ce 

et< 

ae  +  bf 

df 

ef 

a' -a     1 

b'  -b     1 

c'-c     1 

a'            (r  - 

ab  +  W 

a^  -((C  +  (^ 

^2  _  ab  +  b-' 

b- 

U^  _  6c  4-  c» 

a-  —  ac  -\ 

-  c 

'     6-'  - 

-  6c  +  c* 

.     c^ 

72 


THEORY  OF  EQUATIONS. 


Art.  48 


a 

a 

a 

a 

-1 

1 

0 

0 

a 

h 

b 

b 

0 

-1 

1 

0 

a 

h 

c 

c 

0 

0 

-1 

1 

a 

h 

c 

d 

1 

1 

1 

-1 

Solve,  by  means  of  determinants,  the  following  equations : 


63. 


64. 


65. 


67. 


68. 


3  a; +  5  3/ =  17 
2  a; +  3?/ =  11 

4:X  +  7y-10=   0 
7x  —  4:y-{-l=    0 


3x-4:y  +  2z=      11 

2x  +  3y^3z  =  -l  1 

Ans. 

1,  2,  3. 

5x  —  5y  +  iz=      7J 

4:X-7y+    2  =  161 

3.^+    y-2z  =  10\ 

Ahs. 

5,  1,  3. 

5x-6y-3z  =  10) 

5x-4.z=     42 

3z  +  5y=       1 

A 

ns.  6, 

2,  -3. 

Ay-3x  =  -10 

4x  +  7y  -\-3z  —  2iv=    9 

2x—    y-4:Z  + 3 IV  =  13 

3x  +  2y  —  7z  —  4:iv=    2 

Ans 

1,3, 

-1,3. 

5x  —  3y+    2  +5  10  =  13 

69. 


3  .T  +  2  ?/  +  4  z  —  ?c'  =  13 
5x+  y  —  z  +  2w=  9 
2  a;  +  3  //  -  7  2  +  3  ?o  =  14 
4:X  —  Ay-{-3z  —  5io=    4 


^ns.  2,  4,  -1,-3. 


Art. 


MTSCEL  L  A  NEO  US   EX  A  MPL  ES. 


70.  What  relation  must  exist  between  a,  h,  c,  d  if  the  equa- 
tions 

ax  +  by-{-  cz  +  d=  0, 

bx  +  ay  +  dz  +  c  =  0, 

ax+  cy+  bz  +  d=  0, 

cx  +  ay  +  dz+b=  0, 
be  simultaneously  true  ? 

71.  Test  the  consistency  of  the  system 

x  +  10y  +  Uz=    3 
2x-    6y+    7z=    S 
x  +  12y  +  nz=    4 
X-    3y+    2z  =  12) 

72.  Test  the  consistency  of  the  system 

2x-3y  +  10z=  4 

x  +  4:y-    82=  2 

3x+    y+    2z=  6 

4.'i;  +  5?/+       z=  8 

73.  Solve  the  homogeneous  equations 

x  +  2y  +  3z=  0 
2x  +  3y-\-4.z=  0 
3a;  +  4y  +  52;=    0 

74.  A  skew-symmetric  determinant  of  odd  order  vanishes. 
For   any  skew-symmetric  determinant,  A  (see   Art.   48)   is 

unaltered  by  changing  the  columns  into  rows,  and  then  diang- 
ing  the  signs  of  all  the  rows.     But  when  the  order  of  the 
determinant  is  odd,  this  process  ought  to  change  the  sign  of 
A ;  hence  A  must  in  this  case  vanish.     Fur  example, 
0      a     6 

-a      0     c    =0 

-b   -c     0 


74 


THEORY   OF  EQUATIONS. 


Art.  48 


We  give  as  our  last  example  a  special  determinant  as  the 
product  of  differences.  Exercises  7  and  8,  after  Art.  21,  have 
afforded  examples  of  the  resolution  of  this  particular  form  of 
a  determinant,  of  which  we  now  consider  the  general  case. 

75.  Take  any  n  quantities,  a,  b,  c,  •••,  I',  I,  and  form  a  de- 
terminant containing  as  rows  (or  columns)  the  powers  of  these 
quantities  from  0  to  n  —  1 ;  thus  : 


1 

1 

1 

...    1 

1 

a 

b 

c 

•  •     A; 

I 

cC- 

b' 

c' 

...     k- 

I- 

a' 

W 

c« 

...     A;^ 

P 

^•"-^   I" 


This  determinant  possesses  the  property  of  vanishing  when 
any  two  of  the  n  numbers  are  equal,  for  example,  if  we  put : 

a  =  b,  a  =  c,  ...,  a  =  I,  b  =  c,  b  =  d,  etc., 

since  then  two  columns  become  identical.  It  results  that  A 
ought  to  contain  as  factors  all  the  differences  which  can  be 
formed  with  the  series 

a,  b,  c,  ...  k,  I, 


ill  subtracting  from  each  letter  all  the  letters  that  follow  it. 
The  product  P  of  these  differences  would  be 

P  =  (a  -  b)  (a  -  c)  (a  -  r?)  •  •  •  (a  -  k)  (a  -  I) 

(b-c)(b-d)-'(b-k)(b-l) 

(c-d)--(c-k)(c-l) 


(h-k)(h-I) 
(k-T). 


Art.  48  MISCELLANEOUS    EXAMPLES.  75 

The  detenninaut  A  is  equal  to  P  iu  absohite  value.     For 
the  degree  of  A  with  respect  to  a,  b,  c,  •••,  I,  is  equal  to 

l  +  2+3  +  ...  +  (n-l)  =  ?^^5^ 

as  we  see  from  its  principal  term ;  this  is  also  the  degree  of  P, 

which  embraces  "  !^  ~,  ^  diiferences ;  therefore  P  and  A  can 

differ  from  each  other  only  by  a  numerical  factor.  Finally,  to 
determine  this  factor,  we  remark  that  the  principal  term  of  A 
is  the  expression 

1  ■b-c''Cp---k"---l"-\ 

The  corresponding  term  of  the  product  P,  obtained  in  con- 
sidering the  columns,  we  find  to  be 

(- 1)6 .  (-  i)V .  (- i)\p ...  (-  i)''--r-2 .  (-  iy-H"-\ 

This  has  for  coefficient 

"C-D 
(_  l)'+2+3+-+('»-l)  _  /        1)      2 

Therefore,  we  have 

A  =  ±P, 

according  as  Ill^Lp — I  is  even  or  odd. 


PART   IL  — THEORY   OF   EQUATIONS. 


INTRODUCTION. 

Historical  Note.  While  we  cannot,  in  this  brief  notice,  go  back  to  the 
beginnings  of  algebra,  a  few  historical  notes  may  prove  of  interest  to  the 
reader.*  The  first  comprehensive  algebra  was  published  in  1494  by  Lucas 
Pacioli,  an  Italian  mathematician.  Scipio  Ferro  (Professor  of  Mathematics 
at  Bologna  from  149()  to  1525)  first  solved  a  cubic  equation  of  the  form 
a:^  +  mx  =  n.    His  method  is  not  known. 

A  second  solution  of  cubics  was  given  by  Nicolo,  called  Tartaglia  (150(5- 
1557).  This  solution,  known  as  Cardan's  Solution,  was  stolen  by  Hieronimo 
Cardano  (1501-1570)  and  published  in  1545  in  Cardan's  Ars  Magna.  Ferrari 
(a  pupil  of  Cardan's)  discovered  a  general  solution  of  bi-quadratic  equations, 
which  was  also  published  in  the  Ars  Ma(/)ia,  a  work  far  in  advance  of  any 
algebra  previously  printed.  About  the  middle  of  the  sixteenth  century  nega- 
tive roots  were  receiving  considerable  attention,  but  it  seems  impossible  to  say 
who  first  fully  comprehended  them.  Bombelli,  in  his  algebra  published  in 
1572,  opened  the  way  to  the  recognition  of  imaginary  I'oots.  Here,  too,  prog- 
ress was  slow.  Michael  Stifel  was  the  greatest  German  algebraist  of  the  six- 
teenth century.  Vieta  (1540-1(;03),  the  most  eminent  French  mathematician 
of  the  sixteenth  century,  enriched  algebra  by  innovations  in  notation,  and  by 
numerous  discoveries  in  the  Theory  of  Equations.  Thomas  Harriot  (1500-1621), 
of  England,  made  further  improvement  in  notation,  and  did  much  to  establish 
the  Theory  of  Equations  on  a  scientific  basis.  After  this  it  was  enriched  by 
the  fruitful  discoveries  of  Descartes,  Newton,  Lagrange,  Argand,  Gauss,  Abel, 
Hermite,  Kronecker,  Cayley,  Sylvester,  and  others.  The  solution  of  numeri- 
cal equations  was  particularly  advanced  by  Fourier,  Budau,  Horuer,  and 
Sturm. 

There  ai'e  many  text-books  in  which  the  subject  is  discussed,  among  them 
we  may  mention  :  Burnside  and  Panton's  Theory  of  Equations ;  Todhunter's 

*  An  excellent  history  of  mathematics,  and  perhaps  the  one  most  easily  accessible  to 
the  reader,  is  Gajori's  A  IlUtorij  of  MalhemuUca,  Macmillan  &  Co.     Interesting  historical 
notes  may  be  found  in  Fine's  The  yumher-Sijsteni  of  Algebra. 
76 


Art.  49  PA  R  T    II.  —  IN  TR 01)  UCTION. 


(  i 


An  Elementary  Treatise  on  the  Tliconj  of  Equatiunx;  Serret's  Coitrs  d' Al- 
gebra Sap^rieure;  Carnoy's  fours  d'Algebre  Superieure ;  Bieriuaiurs  Kle- 
mente  der  Iloheren  Matheniatik;  Matthiessen's  Grundzdije  der  Antiken  nnd 
Modernen  Algebra  der  Litteralen  Gleichungen;*  Petersen's  Algebraisehe 
Gleichungen. 

49.  In  elementary  algebra  the  student  has  solved  equations 
of  the  iirst  and  second  degrees,  and  has  become  somewhat 
familiar  with  the  meaning  of  the  word  root  as  applied  to  an 
equation ;  and  some  of  the  definitions  given  in  these  pages,  as 
well  as  some  of  the  processes  described  and  employed,  will  not 
be  entirely  new  to  him.     Let  us  consider  the  theorem : 

An  integral  equation  of  the  first  degree  in  one  unknon-n  has 
one  and  only  one  solution. 

For  example,  take  the  equation 

a.v  +  h  =  0 (1) 

One  solution  of  this  is  .r  =  — -.     To  prove  that  this  is  the 
a 
only  root,  let  us  suppose  that  there  are  two  distinct  solutions, 
X  —  u,  and  x  =  /3,  of  (1).     Then  we  must  have 

aa  4-6  =  0, 

ap  +  h  =  0. 

From  these,  by  subtraction,  we  derive 

a(«  -fi)  =  0. 

Now,  by  hypothesis,  a  is  not  =  0,  therefore  we  must  have 
a  — (3  =  0,  that  is,  «  =  )8;  in  other  words,  the  two  solutions 
are  not  distinct.  Hence  there  is  only  one  root,  and  it  is  a 
function  of  the  coefficients.t 

y  *  Matthiessen  develops  the  subject  historipally,  and  on  padres  0(V4-l(ioi  may  be  r.miiil  a 
very  extended  bibliograidiical  list. 

t  .\s  is  well  known,  the  constant  term  b  is  called  a  coefficient,  and  it  is  the  cotllblent 
otafi. 


78  THEORY  OF  EQUATIONS.      ■  Art.  49 

The  quadratic  equation 

ax-  +  bx  +  c  =  0 (2) 

has  two  roots,  namely, 


h  +Vb-  —  'iac  -,     —  b  —  V6-  —  4ac, 


2  a 

and  with  respect  to  these  roots,  we  know  that  their  sum  is 

b  c 
,  and  their  product  is  -;  that  is,  their  sum  is  equal  to  the 

u'  ^  a 

coefficient  of  the  second  term  of  the  equation 

a        a 

Avith  its  sign  changed,  and  their  product  is  equal  to  the  last 
term  of  this  equation.  Thus  the  student  has  seen  that  the 
root  of  an  equation  of  the  first  or  second  degree  may  be  ex- 
pressed in  terms  of  its  coefficients. 

The  general  object  of  this  treatise  is  to  establish  results 
with  respect  to  equations  of  a  higher  degree  than  the  second, 
similar  to  those  that  have  been  established  in  elementary 
algebra  respecting  equations  of  the  second  degree.  In  fact, 
the  science  of  the  Theory  of  Equations  seeks  to  discover  gen- 
eral methods  for  the  solution  of  equations  of  any  degree.  The 
limitations  to  this  search  will  appear  later  (see  Art.  53). 

50.  Definitions.  Any  algebraic  expression  that  depends  upon 
any  quantity  as  x  for  its  value  is  said  to  be  a  function  of  x. 
Thus  3  a;-  —  4  a:  +  16  is  a  function  of  x,  so  also  is  V«"  —  x'. 

An  algebraic  function  involves  the  operations  of  addition, 
subtraction,  multiplication,  and  division  applied  only  a  finite 
number  of  times.*  All  other  functions  are  called  transcendental 
functions,  such  as  logarithmic,  exjjonential,  trigonometric,  and  in- 

*  Thi.s  of  course  includes  involution  .and  evolution  with  constant  exponents.  See 
Appendix  A. 


Art.  51  PART    II. — IM'liODL  (Jloy.  T«J 

verse  trigonometric.  In  this  work,  when  we  use  the  word  fuuc- 
tion,  we  mean  an  algebraic  function,  unless  it  is  expressly  stated 
or  shown  by  the  form  that  the  function  is  transcendental. 

A  function  of  x  is,  for  brevity,  represented  by  F{x),  f(x), 
<f>  (x),  or  some  such  symbol.     Thus,  for  example, 

F(x)  =  3  .ir  -  4  .i-  +  16,    f(x)  =  a  log  .v,     4>  (x)  =  sin  3  x. 

A  rational  fund  ion  of  a  quantity  is  one  that  contains  the 
quantity  in  a  rational  form  only ;  that  is,  a  form  free  from 
fractional  indices  or  radical  signs. 

An  integral  function  of  a  quantity  is  a  rational  function  in 
which  the  quantity  enters  in  an  integral  form  only ;  that  is, 
never  in  the  denominator  of  a  fraction. 

A  rational  integral  function  of  x,  as  discussed  here,  is  one 
that  can  be  put  in  the  form 

ax'^  +  6.«"~^  +  c.i-"~-  +  •••  +  'kx  +  /, 

in  which  n  is  a  positive  whole  number,  and  a,h,  c  •••I  denote 
any  real  expressions  not  containing  x.  It  will  be  observed 
that  the  coefficients  may  be  irrational  or  fractional. 

Algebraic  symbols  are  numerals,  letters  of  the  alphabet,  or 
conventional  signs  to  denote  certain  operations  or  relations, 
such  as  — ,  +,  X,  H-,  =,  >,  or  <,  etc. 

An  algebraic  expression  is  any  combination  of  algebraic  sym- 
bols which  represents  a  quantity. 

A  term  is  an  expression  whose  parts  are  not  separated  by 

the  signs  +  or  — ,  as  4  a^,  3  ahc,  or  — 

A  monomial  is  an  algebraic  expression  of  one  term  ;  a  itobj- 
nomial  is  one  of  two  or  more  terms. 

51.  An  identical  equation  is  the  statement  of  equality  be- 
tween mathematical  expressions  which  are  either  the  same, 
initially,  or  become  the  same  by  the  apidication  to  one  or  Ixith 
of  the  allowable  mathematical  operations;  for  example, 


80  THEORY  OF  I^QUATIONS.  Art.  51 

a^  _  ^2  _  (^_^  _  y-^  (^j.  ^  yj^  sij;i  2  yl  =  2  sin  ^4  cos  A, 

are  identical  equations. 

If  one  algebraic  expression  containing  x  is  equal,  for  certain 
values  of  a;,  to  another  differently  constituted,  the  equality  thus 
formed  is  called  an  equation  of  condition.  Whenever  an  equa- 
tion of  condition  is  meant,  we  shall  use  the  single  word  equation. 

An  equation,  then,  is  the  statement  of  an  equality,  which  is 
true  only  for  certain  values  of  the  unknown  quantity. 

Any  value  of  x  which  satisfies  this  equation  is  called  a  root 
of  the  equation.  The  determination  of  all  possible  roots  con- 
stitutes the,  complete  solution  of  the  equation. 

By  bringing  all  the  terms  to  one  side,  we  may  obviously 
arrange  any  equation  according  to  descending  powers  of  x  in 
the  following  way  : 

ao^;"  +  ttiX"'^  +  aa"-"^  +  a.x"-"  + \-  a„_iX  -f-  a„  =  0    .     (1) 

An  equation  is  not  altered  if  all  of  its  terms  be  divided  by 
any  quantity.  Dividing  (1)  by  a^,  and  thus  making  the  co- 
efficient of  x"  equal  to  unity,  it  may  be  written  in  the  form : 

a;"  +  p^uf-'  +  pox^'--  H h  p„_iX  +  p„  =  0      .     .     (2) 

The  highest  power  of  x  in  this  equation  being  n,  it  is  said  to 
be  an  equation  of  the  nth  degree  in  x. 

An  equation  is  complete  when  it  contains  terms  involving  x 
in  all  its  powers  from  n  to  0,  and  incomplete  when  some  of  the 
terms  are  absent ;  that  is,  when  some  of  the  coefficients,  aj,  a,, 
tts,  etc.,  are  equal  to  zero. 

The  term  a,„  which  does  not  contain  x,  is  called  the  ahmlnte 
term. 

52.  A  numerical  equation  is  an  equation  in  which  the  co- 
efficients are  represented  by  figures  only ;  a  literal  equation  is 
one  in  which  the  coefficients  are  represented  wholly  or  in  part 
by  letters. 


^t.  53  PART  II. — lyrnoDUCTION.  81 

A  linear  equation  is  one  of  tlie  lirst  degree. 

A  quadratic  equation  is  one  of  the  second  degree. 

A  cubic  equation  is  one  of  the  third  degree. 

A  biquadratic,  or  quartic  equation  is  one  of  the  fourth  degree. 

A  quintic  equation  is  one  of  the  fifth  degree. 

A  sextic  equation  is  one  of  the  sixth  degree. 

Equations  above  the  second  degree  are  called  higher  equiUions. 

53.  In  both  mathematical  and  ph3'sical  researches,  we 
frequently  meet  with  problems  that  involve  the  solution  of 
equations. 

As  the  equations  thus  met  with  are  often  liigher  than  the 
second  degree,  it  becomes  a  matter  of  importance  to  find,  if 
possible,  some  general  method  for  the  solution  of  higher  e<iua- 
tions.  In  the  case  Avhere  the  coefficients  of  an  ecpiation  are 
given  numbers,  very  great  progress  has  been  made  in  discover- 
ing methods  for  the  determination  of  the  numerical  values  of 
the  roots ;  but  the  same  progress  has  not  been  made  in  the 
general  solution  of  equations  whose  coefficients  are  letters. 

We  have  seen  (Art.  49)  that  there  is  a  general  algebraic 
solution  of  literal  equations  of  the  second  degree.  Similar 
formulas  (subject  to  some  limitations)  have  been  discovered 
for  the  solution  of  equations  of  the  third  and  fourth  degrees. 

Many  attempts  were  made  to  reduce  similar  general  formidas 
for  equations  of  the  fifth  and  higher  degrees,  but  without  suc- 
cess ;  and,  finally,  in  1824,  Abel*  proved  the  impossibility  of 
solving  by  radicals  an  algebraic  ecpiation  of  the  fifth  degree, 
or,  in  general,  of  any  degree  higher  than  the  fourth.  This 
important  proof  was  published  V)}'  Abel  in  l.SLfd.f  In  modern 
form  it  may  be  be  found  in  Biermann.t  Serret§  gives  a 
simpler  proof  by  Wantzel. 

*  Niels  Ileurick  Abel  tlsii.'-ls-.'n),  „f  Xorway. 

t  Mihnoire  xur  leu  EqintlioiiH  A/i/rhrii/iien :  Christlania,  t*26.  .M.«o  In  (^rftlr'n  .loiir- 
mil.  M.  I.,  lS-26. 

X  EUmente  der  Hoheren  ilnthfmatik.  %  Courn  il' Alyebre  Sup^r >'■•'■■■•    '' !  I 


CHAPTER   IV 

COMPLEX   NUMBERS. 

54.  In  the  solution  of  quadratic  equations,  the  student  has 
frequently  met  with  the  square  root  of  a  negative  quantity. 
Such  a  number  is  said  to  be  imaginary  or  unreal,  for  the 
square  of  no  real  quantity  is  negative.  The  imaginary  unit 
V—  i  is  denoted  for  brevity  by  i,  and  integral  powers  of  i 
beyond  the  first  can  always  be  reduced  by  the  relation  /-  =  —  1. 
All  the  operations  that  we  perform  on  the  unit  i  must,  then, 
be  subject  to  this  definition,  i^  =  —  1,  and  to  the  general  laws 
of  algebra.  For  example,  yi^iy,  yi+y'i=(y-\-y')i=i{y  +  y'), 
etc.,  exactly  as  if  i  were  a  real  quantity. 

55.  If  we  combine,  by  addition,  any  real  quantity  a  with  a 
purely  imaginary  quantity  bi,  there  arises  a  mixed  qiumtity 
a  -f  bi,  a  form  frequently  met  with. 

Such  an  expression,  consisting  of  a  positive  or  negative  real 
units  and  b  positive  or  negative  imaginary  units,  is  called  a 
corapiex  number,  or  quantity.  (Throughout  this  book  we  make 
no  distinction  between  the  words  "number"  and  '* quantity.") 

lieal  and  purely  imaginary  numbers  are  both  included  in 
the  expression  a  +  ib,  the  former  being  obtained  when  6  =  0, 
and  the  latter  when  a  =  0. 

Of  course,  in  such  expressions,  a  and  b  are  considered  real. 

56.  The  successive  powers  of  i  are  periodic.     We  have : 

i^  =  /,  i^  =  —  Ij         i^  =  i^ .  i  =  —  i^ 

i*  =  i- .  r  =  -f  1,  i^  =  i^  •  i  —  -\-  i,  etc. 

82 


Art.  60  COMPLEX  yUMIlEUS.  83 

Beginning  with  the  fifth  power,  nil  the  results  repeat  them- 
selves in  the  same  order.  There  are  only  four  different  values, 
namely  :   +  /,  —  1,  —  i,  +  1- 

57.  If  X  +  ill  —  0,  then  must  x=0,  y  =  0.  Otherwise  we 
should  have  x=  -  i>/;  but  x  is  real  by  hyputhesis,  and  hence 
X  cannot  equal  —  iij,  which  is  imaginary. 

58.  If  X  +  ii/  =  a  +  ib,  then  x  =  a,  y  =  h.  Otherwise  we 
should  have  x  —  a=  i(b  —  y),  which  cannot  be,  since  x  —  a  is 
real. 

59.  The  alr/ebmic  sum  of  any  number  of  com2)lex  q  nan  titles 
is  a  complex  quantity. 

Suppose  we  have,  say,  three  complex  luimbers,  .r,  -f  y,/, 
X2-\-yJ,  Xs+y^i,  then  (xi+yii)  +  (xo-tyJ)-(x3+y/)  =  (Xi  +  .r.,-.rs) 
+  {yi  +  y2—  yi)h  by  the  laws  of  algebra  already  established. 
But  a?!  +  x'2  —  .1-3  and  ?/i  +  ^^2  —  Vi  are  real,  since  x^,  x.>,  x-j,  y,,  y.^ 
ys  are  real.  Hence  (Xi  +  X2  —  x^)  -f  (?/i  4-  ?/,  —  .Vs)*"  is  a  complex 
number.  The  conclusion  obviously  holds,  however  many  terms 
there  may  be  in  the  algebraic  sum.  For  special  case  where 
the  sum  is  real  see  Art.  04. 

60.  The  2-)roduct  of  any  number  of  complex  numbers  is  a 
complex  number. 

Consider  the  product  of  two  complex  numbers,  .r,  +  yii  and 
X2  +  yoi-     We  have 

(^1  +  z/iO  (-^'2  +  yJ)  =  -^'la.'!.  +  yiy/-  +  ^\yJ  +  a-.^/J- 

Hence,  bearing  in  mind  the  definition  of  /,  we  have 

(^i  +  .'/lO  (-'^2  +  y-/)  =  (^i->'*2  -  yi.V2)  +  (-''LVi  +  av/,)', 

Avhich  proves  that  the  product  of  two  comi)lex  numbers  is  a 
complex  number.  The  proposition  is  easily  extended  to  a 
product  of  three  or  more  complex  numbers.  For  special  ease 
where  the  product  is  real  see  Art.  04. 


L    )C   L   :^   Q 


84  THEORY  OF  EQUATIONS.  Art.  61 

61.    TJie  quotient  oftico  complex  mmibers  is  a  complex  number. 
We  have 

X2  +  yd         xi  -  (jj4f 


{x^x.,  +  y^y.^ 

-(^i2 

h.  -  x&iY 

xi 

+  2/2^ 

XxX2  +  y,yo 

xi  +  yi 

+  yir 

which  proves  the  proposition. 

Cor.  I.  Since  every  rational  function  involves  only  the 
operations  of  addition,  subtraction,  multiplication,  and  divi- 
sion,- it  follows  from  the  above  theorems  that  every  rational 
function  of  two  or  more  complex  numbers  can  be  reduced  to  a 
complex  number. 

Cor.  II.  If  f{x  -f-  yi)  be  any  integrcd  function  of  x-\-  yi,  having 
all  its  coefficients  real,  and  if 

f{x  +  yi)  =  P  +  Qi, 

then  f{x  -  yi)  =  P-  Qi, 

ivhere  P  and  Q  are  real. 

For  it  is  obvious  that  P  can  contain  only  even  powers  of  y, 
and  Q  only  odd  powers  of  y.  If,  therefore,  we  change  the  sign 
of  y,  P  will  remain  unaltered,  and  Q  will  simply  change  its 
sign.     Hence  the  theorem. 

Cor.  III.  If  <t>(x  -{-  yi)  be  any  rational  function  of  x  +  yi, 
having  all  its  coefficients  real,  and  if 

<}>(^x  +  yi)^X+Yi, 
then  <f>{x  —  yi)  =  X  —  Yi. 


Art.  64  COMPLEX    ^'rMnEll.'i.  85 

EXAMPLES. 

1.  3(3  +  2  0  -  2(2  -  3  0  +  (<•»  +  8  /)  =  11+20  i. 

2.  (2 +  3  0(2 -30(3 -5/)  =  (4 +  1)^(3-50  =  30 -05/. 

3    3 +  5/^  (3 +  5  0(2 +  30^      0       10. 
•    2-3i  4  +  9  13      13'" 

4.    {x  +  yiy  =  (.«••  -  6  ayy'  +  ^^)  +  (4  x'y  -  4  .r/)  /. 

62.  Two  complex  numbers  which  differ  only  in  tlie  si«,Mi  of 
their  imaginary  part  ai-e  said  to  be  co»J"gote. 

Thus  -  3  -  2  ?:  and  -  3  +  2 « ;  -  4 1  and  +  4  t ;  x -\-  >/i  and 
X  —  yi,  are  C(jnjugate. 

The  student  has  met  with  conjugate  imaginaries  in  tlie 
solution  of  quadratic  equations,  where  if  one  root  is  imaginary, 
the  other  is  also  imaginary,  and  is  conjugate  to  the  first. 

63.  If  a  +  ib  is  a  root  of  an  cdgebraic  equation,  then  also  is 
a  —  ib  a  root  of  the  same  equation. 

For,  let  f(x)  =  0  be  the  equation.  If  a  +  ib  is  a  root,  we 
must  have /(a  +  ib)  =  0.     This  may  be  written 

f(a  +  ib)  =  0=  P+iQ  =  0; 

and  this  requires  P=  0,  Q  =  0  (Art.  57).     Hence  P-iQ  =  0, 
and  f(a  —  ib)  =  P  —  iQ=z{)-^  hence  a  —  ib  is  a  root  of  f(x)  =  0. 

64.  The  sum  of  the  conjugate  imaginaries,  x  +  ///,  x  —  iy,  is 
the  real  quantity  2  x ;  their  difference  is  the  pure  imaginary 
2  iy. 

Their  product  x-  +  y'  is  called  the  norm  of  either  of  them. 

noi-ni  (x  +  iy)  =  norm  (x  —  iy)  =  ar  +  y-. 


86  THEORY  OF  EQUATIONS.  Art.  64 

The  modulus  of  a  complex  quantity  is  the  positive  square 
root  of  the  norm.     Thus,  employing  the  usual  symbol, 


mod  {x  +  iy)  =  Va;'^  +  y'\ 


mod  (x  —  iy)  =  VX'  +  y-. 

Rein.  When  y  =  0,  that  is,  if  the  complex  number  be  wholly 
real,  then  the  modulus  reduces  to  +Va;-,  or  x,  that  is  simply 
the  numerical  value  of  x.     For  example, 


mod  (-  3)  =  +  V(-  3f  =  +  3,  mod  (+  5)  =  +  5. 

EXAMPLES. 

norm  (-  3  +  4  0  =  (-  3)^  +  (4)'  =  25. 
norm  (4  —  5  i)  =  41. 
mod(-3  +  4i)  =  5. 

mod  (2-5  0  =  V29. 
mod  (1  +     i)  =  V2. 
mod  (6  +  8  0  =  10. 

65.  If  a  complex  number  vanish,  its  modulus  vanishes;  and 
conversely,  if  the  modxdus  vanish,  the  complex  number  vanishes. 

For,  if  X  +  yi  =  0,  then  a;  =  0,  and  y  =  0. 

Hence  Vx-  +  if  =  0. 

Again,  if  Vx^  -{-y-  =  0^  then  x"^  -\-y"  =  0,  hence,  since  x  and 
y  are  real,  a;  =  0  and  y  —  0. 

66.  If  tu:o  complex  numbers  are  equal,  their  modidi  are  eqmd. 
For,  if  X  +  yi  =  x'  +  y'i,  then  x  =  x',  y  =  y' ; 

hence  V.r^  +  y-  =  Va;'-  +  y''\ 

The  converse  is  obviously  not  true. 


Art.  6/ 


COMPLEX  NUMBERS. 


67.   Graphic  Representation.  —  Argand's  Diagram.*     "We  shall 

consider  now  tlie  giai)liu'  method  of  reprcsLMiting  complex 
numbers  originally  suggested  by  Argand. 

We  have  seen  that  the  usual  representation  of  positive  or 
negative  quantities  is  by  means  of  distances  measured  along  a 
straight  line,  positive  quantities  being  represented  by  distances 
measured  to  the  right,  negative  quantities  by  distances  to  the 
left.  For  some  reasons  it  is  best  to  say  that  positive  quanti- 
ties are  represented  by  distances  measured  to  the  right,  and 
that  the  effect  of  multiplying  any  quantity  by  —  1  is  to 
reverse  the  direction;  that  is,  if  the  quantity  is  multii)lied 
twice  by  i,  the  direction  is  reversed. 

If  now  the  factor  /-,  or  i  •  i,  changes  the  direction  by  180°, 
then  it  seems  natural  to  consider  *  a  factor  that  changes  the 
direction  by  90°.  It  is  customary,  to  say  that  the  effect  of 
multiplying  by  i  is  to  tuni  the  line  through  an  angle  of  90°  in 
the  positive  direction  (counter-clockwise).  It  is  evident  that 
the  repetition  of  the  operation  of  using  i  once  as  a  factor, 
reverses  the  direction. 

jS^ow,  let  XOX',  YO  Y'  be  two  rectangular  axes.     We  shall 


Fig.l 


X'- 


*  So  called  because  to  Arpand  is  due  the  credit  of  first  gMng  this  peometriral  construc- 
tion in  his  KMmi  mir  une  mtniieri'  de  reprenenier  lex  quauliliH  iinngintiirm  tlnnn  let 

coustruetions  gevmHriqueH  (l^)ii).     SfC  ChrysUl's  Al(/ebrti,  Vol.  I.,  p    -•»"      < "> 

Appendix  B. 


88  THEORY  OF  EQUATIONS.  Art.  67 

call  XOX'  the  axis  of  real  quantity,  YO  Y'  the  axis  of  purely 
imaginary  quantity. 

To  represent  the  complex  number  x  +  iy,  we  lay  off  on  the 
a>axis  tlie  distance  OM  —  x,  and  on  MP,  perpendicular  to  the 
X-axis,  the  distance  3IP  =  y. 

Thus  the  poiut  P  is  definitely  located  by  the  quantity  x  -{-  iy. 
The  distance   OP  =r  =  Vx-  +  y-  =  mod  (x  -j-  iy),  and  we  have 

cos  MOP  =  cos  ^  =  -,  sin  0  =  t 
r  r 

Hence  the  expression  x  +  iy  may  be  written  in  the  form 

?*(cos  0  +  i  sin  6). 

The  quantity  r  is  called  the  modulus*  and  the  angle  0  the 
argument  of  the  complex  nnmber  x  +  iy.   • 

The  modulus  and  argument  of  a;  +  iy  are  for  brevity  repre- 
sented by  the  notation 

mod  (x  +  iy),  arg  {x  +  iy). 

Example  :  To  write  3  +  4  i  in  the  trigonometric  form 
r(cos  6  +  i  sin  6),  we  have 

r  =  V32  +  4-  =  5,   cos  ^  =  f ,    sin  6  =  |, 
and  .-.  3-f  4i  =  5(f +  i4). 

Cor.  Of  course  x—iy,  or  r(cos  6  —  i  sin  0),  represents  the 
point  P',  the  y  in  this  case  being  measured  downward  because 
it  is  negative.  If  the  argnment  of  x  +  iy  is  $,  the  argument  ^i 
x  —  iy  is  2  7r  — ^,  or  we  may  say  that  two  conjugate  numbers 
have  the  same  projection  on  the  a;-axis. 

68.  The  Exponential  Form  of  jr  +  //.  The  following  develop- 
ments for  cos  $,  sin  0,  and  e'',  which  are  deduced  in  works  on 
trigonometry  and  elementary  calculus,  are  supposed  to  be 
known : 

*  fiennan  writers  uso  " absolute  value"  instead  of  "modulus,"  and  denote  it  by  tlie 
symbol  |  x  +  it/ 1.    Thus  '•^j-^  +  //"  =  |  ^r  +  /(/  |  =  absolute  value  of  the  complex  number  ir  +  iy. 
So  also  6  is  often  called  the  "amplitude"  instead  of  the  "argument." 


Alt.  08  COMPLEX  ^UMliERS.  89 

e'  =  l+a;  +  — +  —  +  —  +•-, 
2!3!4! 

008^  =  1--^  +  -^--^+..., 


o  .      o  .       <  ! 
From  the  last  two  we  have 

cos  e  +  i  sill  ^  =  1  +  ^^  _  -^  _  /-^  +  -^  +  /il . 

2!       3!      4!       5! 

If  we  define  a  function  e'^  by  the  series 

2\         J I         4  !        5 ! 

2!       3!      4!       5!^ 

which  is  entirely  analogous  to  the  form  for  e',  where  x  is  real, 
then  we  have 

e''«  =  cos  ^  +  i  bin  0, 
and,  consequently, 

x+  iy  =  r(cos  ^  +  ?'  sin  9)  =  re'*. 
Similarly,       x*  —  iy  =  9-(cos  ^  —  1  sin  ^)  =  re~*^. 

Cor.  I.     The  following  fornmlce  are  sometimes  useful : 
e'"  =  cos  IT  +  i  sin  tt  =  —  1, 
e-'>  =  cos  TT  —  ?■  sin  tt  =  —  1, 

.-rr  TT     ,      .       •       TT 

e'2  =  cos  -  +  I  Sin  -  =  i, 


e-'f  =  cos  J  -  i  sin  J  =  -  «• 


90  THEORY   OF  EQUATIONS.  Art.  68 

Cor.  II.     \i  6  =  ^,  and  r  =  1,  then  x  +  iy  becomes 


'     '  1  ■  (cos^  +  i  sm~\  =  l  '  i  =  e+'l. 

Hence  e"*"''?  =  i,  is  the  operator  which  turns  the  direction 
through  90°. 

69.  Expressing  e'^,  e'"*  in  their  respective  trigonometric  forms, 
and  performing  the  operations  of  multiplication  and  division, 
we  can  readily  prove  the  relations : 


Hence,  the  function  e'^,  defined  in  the  last  paragraph,  obeys 
the  same  laws  of  multiplication  and  division  as  the  function  e'', 
where  a;  is  real. 

70.  De  Moivre's  Theorem.  First,  for  n  a  positive  whole 
number. 

If  in  the  equation 

(x  +  iy)  (a  +  ib)  =  ?jtre'^^+"^,* 

we  let  a  -\-  ib  =  X+  iy,  it  becomes 

(x  +  iyf  =  r'e-  •  2«  =  ,^  (cos  2  ^  +  ?  sin  2  ^) ; 

similarly,    (x  +  iy)"  =  r"  •  e'  •  "*  =  r" (cos  nO  +  i  sin  nO). 

Hence,  for  n  a  positive  whole  number, 

(cos  6  -j-i  sin  6)"  —  cos  n9  +  i  sin  ?i^. 

Second,  for  ?i  a  negative  whole  number. 
We  know  that 

*  Here  a  +  ib  =  we*".  .    d 


Art.  70  COMPLEX  NUMIiEIiS.  91 

■  e"        - 
a  +  to      m 

If  in  this  we  make 

(a  +  lb)  =  (;«  +  ii/)"+^  =  ,-n+ie'(n+i)« 
we  shall  have 

Hence 

[r(cos  0  +  i  sin  ^)]-"  =  r-''[cos  (-  nO)  +  i  sin  (-  «^)]. 

.-.  (cos  ^  +  i  sin  ^)-"  =  cos  n6  —  i  sin  h$. 

Hence  (e'^)"  =  e'"^,  where  u  is  any  positive  or  negative  wliole 
number. 

Third,  n  any  number. 

Suppose  that  $  =  ^,  then  e'^  —  e^j,  and  (e**)'  =  (e'7)'=>»=g'*. 
That  is,  the  tth  power  of  e'r  is  e''i> ;  conversely,  one  of  the  ^th 
roots  of  e'*  must  be  e'Y; 

hence  (cos  ^  +  i'  sin  6f  =  cos  -  +  i  sin  — 

^  t  t 

Finally,  if  s  and  t  are  any  wliolauumbers^we  have 
(e'").  =  e*^  9  =  cos  -  ^  +  i  sin  ^  ^ ; 

but  as  s  and  t  are  any jmmbers,  wJiatfivei-,  -  may  represent  anx. 

rational  or  irrfttiinnil 'Tiunrberj    hence,  when  n  is  any  number 
whatever,  integer,  fractional,  or  liiaLiuim^  we  have 

(e'9)"  =  e"'", 

or  (cos  6  +  i  sin  6y  =  cos  nO  +  t  sin  ud, 

which  is  De  Moivre's  Theorem* 

*  Abraham  de  Moivre  (ir)6T-n54).    The  discovery  of  this  theorem  by  De  Muivre  revohi- 
tionized  analytical  trigonometry. 


92  THEORY  OF  EQUATIONS.      ■  Art.  71 

71.  The  Values  of  (e'^)",  for  integer  value  of  n. 
By  definition,  we  have 

e"'"  =  cos  2  TT  +  t  sin  2  TT  =  1 ; 
hence  e'^  •  e-''"  =  e'^  =  e'(9+2,r)^ 

or,  more  generally, 

gte  _  (_j(e+-2kn)^  where  k  is  any  whole  number  whatever. 

1  1  ,e+gA-r 

Hence  (e'«)"  =  (e-(e+2*>r))n  ^  e'    "  ', 

whence  (e'^Y  =  cos  1+1^  +  ,•  gin  1±1^ 

?i  n 

where  k  may  be  any  whole  number.     While  from  this  equation 

the  number  of  values  of  (e*^)"  is  apparently  infinite,  there  are 

really  only  n  different  values,  for  Avhen  k  has  run  through  the 

1 
numbers  0,  1,  2,  3,  •••,  n  —  1,  the  values  of  (e'^)"  begin  to  repeat 
themselves,  as  may  be  readily  shown. 

72.  Solution  of  the  Equation  a-"  —  1  =  0. 

This  is  a  special  form  of  the  binomial  equation,  the  general 
form  of  such  equations  being  a;"  =  a  +  6v  —  1,  where  a  and  b 
are  real  quantities.     To  find  the  roots  of 


(1) 


we  have 


X"  —  e-"^  =  e" 


Hence    ic  =  e^^^  =  cos  5-^^+1^  +  i  sin  ?iil±l^    .     (2) 
n  n 

Yov  k  =  71  —  1,  we  have 

2mr  ,    .  .    2mr  ^         .   ■    ^ 

X  =  cos h  i  sm =  cos  2  TT  +  ;  sm  2  tt  =  1. 

n  n 

Tlierefore  +  1  is  a  root  of  the  ecpiation,  .r"  =  1. 


Alt.  73  COMPLEX    SUM  HERS.  y^ 

If  11  is  even,  we  may  make  A-  =  ^  —  1,  then  we  have 

7i7r    ,    .    .     Uir  .    . 

»  =  COS h  I  sm  -  -  =  cos  TT  +  I  sin  tt  =  —  1, 

Hence,  if  n  is  even,  both  + 1  and  —1  are  roots  of  x"  =  1. 
But  if  /i  is  odd,  +  1  is  the  only  real  root.    This  is  evident  from 

the  fact  that  for  all  values  of  k,  other  than  "  —  1  for  n  even, 

o     _i_  *?  z-      "^ 
and  n  —  1  for  n  even  or  odd,  sin  "^  ^  "*"  *"        is  not  zero,  and 

II, 
therefore  the  root  is  imaginary. 

73.   Solution  of  the  Equation  jr"  +  l  =  0.     To  find  the  roots  of 
X"  =  -!..., 
Ave  have  a;"=  e''<"+-*'^',  since  e'"' =  —  1. 

Hence     a;  =  e'     »     =  cos  — f- 1  sin  ^!^ 

n  n 

If  n  is  even,  the  roots  are  all  imaginary,  since  no  even  power 

of  a  real  quantity  can  be  negative ;    but  if  n  is  odd,  we  may 

make  A:  =     ~     ;  then  we  find  x  =  cos  tt  -\-  i  sin  tt  =  —  1.     "We 

conclude  that  when  n  is  odd,  there  is  one  and  only  one  real 
root,  —  1. 

EXAMPLES. 

1.  rind  the  cube  roots  of  +,1. 

Here  ar^=l,  and  in  equation  (2),  Art.  72,  A"  may  be  made  equal 
successively  to  0,  1,  2,  while  n  =  3.     We  thus  get  for  the  roots 

X  =  cos  f  TT  +  '  sin  I  TT  =  —  J-  +  ^  V3  i, 
X  =  cos  ^TT  +  i  sin  |  tt  =  —  |  —  ^  V3  i, 
X  —  cos  2  TT  +  /  sin  2  TT  =  4- 1- 

2.  Solve  the  equations  x*  =  1,  and  x*  =  —  1. 


^ 


94  THEORY  OF  EQUATIONS. 

3.  Solve  the  equation  x^  =  1. 
The  roots  are : 

-i(V5+i)+iva<>-2V5)i, 

-i(V5  +  l)-iV(10-2V5)/, 

.^J      KV5-i)-iva^>  +  2V5)i. 

4.  Solve  the  equation  x^"  ~  1. 


74.  Complex  Numbers.  —  Addition.  Let  rectangular  axes  be 
taken  and  a  point  P  representing  a-\-ih',  that  is,  Art.  07, 
OM^cu  PM=b,  and 


4^    c^j    X'    -   ■ 
Art.  73 


(At- 


OP  =  Va-  +  &-  =  ^  =  mod  (a  +  ih),  and  MOP^  a = arg  (a  +  /6). 
F 


rig 


B- 


Let  a  second  complex  number  a'  +  ib'  be  represented  by  the 
poinr  A,  so  that 

0.4  =  mod  (a'  +  (6'),    XOA  =  arg  (a'  +  ib'). 

Now  the  sum  of  these  two  complex  numbers  is 

a  +  ib  -f-  a'  +  ib', 


Art.  75  COMPLEX  Xi'MBEIiS.  95 

which  may  be  written  in  the  form 

a  +  a'  +  i(b  +  b'), 

and  we  observe  that  this    sum   is   represented   by  the   point 
whose  coordinates  are  a  +  a',  b  +  b'. 

To  find  this  point  draw  PP'  parallel  and  equal  to  0A\  since 
PC,  PC  are  equal  to  a',  b',  P'  is  the  required  point,  and  we 
have 

«^ 
0P'=  mod  \  a  +  a'  +  /  {b  +  b')\,  XOP  =  arg  *  a  +  a'+i  {b  +  b')l. 

Therefore,  to  add  two  complex  numbers,  represented  by  the 
points  A  and  P,  Ave  draw  PP*  equal  and  parallel  to  OA ;  then 
P'  represents  the  sum  of  the  two  complex  numbers. 

Since  OP'  is  not  greater  than  OP  +  PP',  it  follows  that  the 
modulus  of  the  sum  of  two  complex  numbers  is  less  than  {or  at 
most  equal  to)  the  suui  of  their  moduli. 

To  add  a  third  complex  number  a"  +  ib",  represented  by 
B,  we  draw  P'P"  parallel  and  equal  to  OB.  Then  P"  repre- 
sents 

a  +  a'  +  a"  +  i{b  +  b'  +  b"), 

which  is  the  sum  of  the  three  given  complex  numbers. 

As  this  mode  of  representation  may  be  extended  to  the 
addition  of  any  number  of  such  quantities,  it  is  evident  that,  in 
general,  the  modulus  of  the  sum  of  aivj  number  ofcouij>le.c  quan- 
tities is  less  than  (or  at  most  equal  to)  the  sum  of  their  moduli. 

75.  Subtraction.  Subtraction  can  be  represented  in  a  similar 
way.  Since  /-*'  represents  the  sum  of  /-*  and  .1,  1*  will  repre- 
sent the  difference  of  P'  and  A.  To  subtract  two  complex 
numbers,  therefore,  we  draw  from  the  point  repr(>senting  the 
minuend  a  line  parallel  and  equal  to  the  line  from  the  origin  to 
the  point  representing  the  subtrahend,  but  in  the  opposite 
direction.     We  join  O  to  the  extremity  of  this  line  to  find  the 


96  THEORY  OF  EQUATIONS.  Alt.  76 

modulus  of  the  point  which  represents  the  difference  of  the 
two  given  complex  numbers. 

76.  Multiplication  and  Division.  The  theorems  of  Arts.  60 
and  01  may  readily  be  proved  by  De  Moivre's  Theorem,  as 
follows : 

To  multiply  the  two  complex  numbers  a  +  ib,  a'  +  ib',  we 
write  them  in  the  form 

(a  +  ib)  =  /A  (cos  «  +  i  sin  a),  a'  +  ib'  =  fjt.'  (cos  o&'  +  i  sin  a'). 
Then 

(a  +  ib)  (a'  +  ib ')  =  ixfx'  \  cos  («  +  «')  +  i  sin  (a  +  «')  I , 

which  proves  that  the  product  of  tico  comjilex  nvmbers  is  a  com- 
plex number,  whose  modulus  is  the  product  ofthetico  moduli,  and 
ichose  argument  is  the  sum.  of  the  two  arguments. 

Similarly,  we  may  prove  that  the  product  of  any  number  of 
complex  quantities  is  a  complex  quantity  whose  modulus  is  the 
product  of  all  the  moduli,  and  whose  argument  is  the  sum  of 
all  the  arguments. 

To  divide  a  +  ib  by  a'  +  ib',  we  have  similarly 

fi±JJL  =  H  I  cos  (a  -  a')  +  /  sin  (a  -  a')  | , 
a'  +  ib'      fji' 

which  proves  that  the  quotient  of  two  complex  nmnbe7-s  is  a  coni- 
plex  number  whose  modulus  is  the  quotient  of  the  two  moduli,  and 
■whose  argument  is  the  difference  of  the  two  arguments. 

Cor.  Similar  theorems  for  involution  and  evolution  are 
derived  at  once  from  De  Moivre's  Theorem.* 

*  From  the  formula 


It  is  evident  that,  in  invohition,  8  increases  by  arithmetical  projrression,  while  r  increases 
by  jreometrical  progression. 


CHAPTER    v.* 

PROPERTIES   OF   POLYNOMIALS. 

77.  Reduction  to  the  Form  f{^x)=0.  Any  rational  integral 
function  of  x,  J\x),  uuiy,  as  we  have  seen,  be  put  in  the  form 

f(x)  =  o,,i-"  +  ciix"-^  +  (uv"-'-  +  OyV"  ■■'  H h  ((„_,.»•  +  f(„. 

Any  equation  in  x  having  rational  coefficients  can  be  trans- 
formed into  an  equation  of  the  form  f(x)  —  0,  as  the  following 
example  will  show. 

Example.     Eeduce  '''^  ~  ^^  =  ^-^  -  to  the  form  Fix)  =  0. 
1+x        ^.i_^2 

Clearing  the  given  equation  of  fractions,  we  obtain 

x^-x  +  2  xi  -  2  x'  =  .r-i  +  1  -  3  -  3  .r, 

or,  multiplying  by  x  to  free  of  negative  exponents, 

x"^  -x''-\-2x^  -2x^  =  1  -2x-3x'    .     .     .     (1) 

To  transform  (1)  into  another  equation  with  integral  ex- 
ponents, put  x  =  ]f,  6  being  the  least  common  multiple  of  the 
denominators  of  the  fractional  exponents  of  .i-.     Thus  we  get 

2y2  +  7/"-2y«  +  2y'-f  2/-1  =0    .     .     .     (2) 

which  is  the  required  form,  the  roots  of  (1)  and  (2)  holding 
the  relation  x  =  ?/'. 

*  In  this  and  subsequent  chapters,  we  shall  consider  mainly  the  r.  al  vrtluos  i.f  x.  and 
shall  not  enter  upon  the  general  discussion  of  the  theory  of  the  complex  variable. 

97 


98  THEORY  OF  EQUATIONS.  Art.  77 

EXAMPLES. 
Keduce  the  following  expressions  to  the  form  f{x)  =  0 : 

1.  -^+2x-^x^-x'=l. 

2.  "^1^  =  2 +  x-\ 
1  +  x^ 

3.  ^4:-5x  =  l-3x^. 


.    4.   ViK^^-5a?  =  Vl  —  2x  —  X. 

5.  (x^  -  3  x^)  (1  -  x)  =  (.'^-2  +  1)  (x-'^  -  2). 

78.  We  shall  now  give  two  theorems  concerning  the  relative 
importance  of  the  terms  of  a  polynomial  when  values  very- 
great  or  very  small  are  assigned  to  x. 

AVriting  the  polynomial  in  the  form 

,  (         rfi  1      a.  1  a„_i    1        a„  1  ) 

(  Cfo  X        «o  ^  f'o    ^  «0  •*'    ) 

it  is  plain  that  its  value  tends  to  become  equal  to  aox"  as  x 
tends  toward  co.  The  following  theorem  will  determine  a 
quantity  such  that  the  substitution  of  this,  or  of  any  greater 
quantity,  for  x  will  have  the  effect  of  making  the  term  affic"  ex- 
ceed the  sum  of  all  the  others.  In  Avhat  follows  we  suppose 
ttu  to  be  positive;  and,  in  general,  in  the  treatment  of  poly- 
nomials and  equations  the  highest  term  is  supposed  to  be 
written  with  the  positive  sign. 

Theorem.     If  rn  the  polynomial 

«o-«"  +  «i-^-""^  +  aiicc"-^  +  •••  +  a„_ia?  -f-  a„   -  d 

the  value  -*-+  1,  or  any  greater  value,  be  substitutecl  for  x,  ivhere 

a,,  is  that  one  of  the  coefficients  a^,  a.,,  Og,  •••  a„  whose  numerical 
value  is  greatest  irrespeciive  of  sign,  the  term  containing  the  high- 
est p)0wer  of  X  will  exceed  the  sum  of  all  the  terms  tvhich  follow. 


Art.  78  PEOPERTIKS   OF  POLYNOMIALS.  99 

The  inequality 

ttox"  >  a^x"-^  +  (M'"--  +  •••  +  o„_i.r  +  a„ 
is  satisfied  by  any  value  of  x  whii-h  makes 

cioX"  >  Ot(a;"-'  +  x"  -  +  ...  +  .r  +  1), 

Avhere  a^  is  the  greatest  among  the  coeftieients  cr,,  a.^,  •••  «„_|.  a„, 
without  regard  to  sign.  Summing  the  geometric  series  -within 
the  brackets,  we  have 

ao.^•"  >  «*^^^-=^  or  X"  >      /'*    ^(x''  -  1), 
x—1  a^,{x  —  l) 

which  is  satisfied  if  aa(x  —  1)  be  >  or  =  o^ ;  that  is, 
a;  >  or  =  -*  +  !. 

This  theorem  is  useful  in  supplying,  when  the  coefficients  of 
the  polynomial  are  given  numbers,  a  number  such  that  when  x 
receives  values  nearer  to  +  oo,  the  polynomial  will  preserve 
constantly  a  positive  sign. 

If  we  change  the  sign  of  x,  the  first  term  will  retain  its 
sign  if  n  be  even,  and  will  become  negative  if  n  be  odd ;  so 
that  the  theorem  also  supplies  a  negative  value  of  x,  such  that 
for  any  value  nearer  to  —  oc,  the  polynomial  will  retain  con- 
stantly a  positive  sign,  if  n  be  even,  and  a  negative  sign,  if  n 
be  odd. 

As  illustrative  of  the  use  of  this  theorem,  consider  the 
polynomial  10  .v^  —  17  x-  -f-  •'"  +  6. 

Here,  substituting  10  for  (/„  and  17  for  a^,  the  test  formula 

becomes 

a;  >  or  =  J-l  +  1, 

or  a;  >  or  =  2.7, 

which  shows  us  that  the  function  10  a^  -  17  x-^  -f  a;  +-6  retains 
positive  values  for  all  positive  values  of  x  greater  than  L'.7, 
and  negative  values  for  all  values  of  x  nearer  to  —  x  than  2.7. 


100  THEORY  OF  EqUATlOyiS.  Art.  79 

79.  We  next  consider  a  theorem  which  shall  enable  ns  to 
determine  what  term  controls  the  sign  of  a  polynomial  when 
the  value  of  x  is  indefinitely  diminished. 

Theorem.     If  in  the  polynomial 

C(,fX"  +  ClyX"-'^  +    •  •  •    +  ttn-l^  +  a« 

the  value  — ^-^ — ,  or  any  smaller  value,  he  substituted  for  x,  where 

(f,,  +   «A 

a„  is  the  greatest  coefficient  exclusive  of  a^,  the  term  a„  ivill  be 
numerically  greater  than  the  sum  of  all  the  others. 

To  prove  this,  let  x  =  -;  then  by  the  theorem  of  Art.  78,  a^ 

y 

being  now  the  greatest  among  the  coefficients  cIq,  ctj,  •••  a„  i, 
without  regard  to  sign,  the  value  ^  +  1,  or  any  greater  value 


of  y, 

will  make 

«„7/"  > 

«»-i2/"~ 

'+«, 

-22/"-'+    •• 

.  +  a^y  +  tto, 

that 

is 

a„  >  a 

J  + 

y 

r 

••+«.A; 

yU 

hence  the  value  - 

a„ 

or  an 

y  less  value  of  x  will 

«n  +  «* 

ff„  >  a„_iX  +  a„_.2X-  +  •••  -{-aQX\ 

Cor.  I.     This  proposition  may  be  stated  as  follows : 

Values  so  small  may  he  assigned  to  x  as  to  make  the  polynomial 

a„_iX  +  a„  oX-  -\-  •••  +  Oyii;" 

less  than  any  assigned  quantity. 

This  statement  of  the  theorem  follows  at  once  from  the  above 
proof,  since  a„  may  be  taken  to  be  the  assigned  quantity. 

Cor.   II.     Another  useful  statement  of  the   theorem  is  as 
ollows : 


Art.  80  PliOPEHTIES   OF  POLYNOMIALS.  IQl'.':' 

Wheii  the  variable  x  receives  a  very  small  value,  the  sign  of  the 
poUjnomial 

a„_ix  +  «„_2a^  H h  ooo;" 

is  the  same  as  the  sign  of  its  first  term  a„_^^x. 

This  is  evident,  if  we  write  the  expression  in  the  form 

^  1  ^',.-1  +  (',1-2-'''  +  •  •  •  +  <'o-'-"~'  \  ■ 

80.  Derived  Functions.  Change  of  form  of  a  pol tinomial  cor- 
respoiidiitg  to  an  increase  or  dimi)iutio)i  of  the  variable. 

We  shall  now  examine  the  form  assumed  by  the  polynomial 
when  X  +  h  is  substituted  for  x.  Here  the  resulting  form  will 
correspond  to  an  increase  or  diminution  of  the  variable  x, 
according  as  h  is  positive  or  negative. 

The  polynon)ial 

/(.r)  =  UoX"  +  Oi.^"-'  +  a.jX"--  +  o..^"-''  -\ h  o„_,.r  -|-  a„  .   (A) 

becomes,  when  x  is  changed  to  x  +  h,  f{x  +  h),  or 

ao{x  +  hy  +  a,(x  +  hy-'  +  0^(0)  +  hy-^  +  •••  +  «„_,(.r  +  h)  -f  a„. 

Expanding  each  term  of  this  expression  by  the  binomial 
theorem,  and  arranging  the  result  according  to  ascending 
powers  of  h,  we  have 

a^x"  +  tti-x'""^  +  a-yX"'^  +  •••  +  a„_2^^  +  a„_ia;  +  a„ 

+/i|«ao.'i^"~^+(n  — l)o,a;"--+(?i— 2)a2.i;"-'H h -",.-?»; +«,.-i  I 

+  JLln(n  -  l)ao.r»-2+  (n  -  l)(n  -  2)a,r"-'  +  -  +  2a„_2| 
+ 

We  observe  that  the  part  of  this  expression  in(lo])endent  of 
h  is  /(.-<;),  and  that  the  successive  coefficients  of  the  different 
powers  of  h  are  functions  of  x  of  degrees  dimiiiisliing  by  unity. 


* .  i bS*  "••**'•'•  *        THEOR  Y  OF  EQ  UA  TIONS.  Art .  80 

We  also  see  that  the  coefficient  of  h  may  be  obtained  from  f(x) 
by  multiplying  each  term  of  f(x)  by  the  exponent  of  x  in  that 
term  and,  diminishing  the  exponent  of  x  by  unity,  the  sign 
being  retained.  ..  The  sum  of  all  the  terms  of  f{x)  treated  in 
this  way  will  constitute  a  polynomial,  one  degree  lower  than 

.m- 

This  polynomial  is  called  the  first  derived  function  of  f{x), 
and  is  usually  represented  by  the  notation  f'{x).  The  co- 
efficient of  -^  is  gotten  from  /'(a-)  in  exactly  the  same  man- 
ner as  f{x)  is  derived  from  f{x),  or  by  the  operation  twice 
performed  ou  f{x).  This  coefficient,  denoted  by  fix),  is 
called  the  second  derived  function.  In  a  similar  way  the  third 
derived  function,  f"'(x),  is  obtained  from  f"(x),  and  so  on;  so 
that  the  expression,  B,  may  be  written  as  follows : 

fix+h)=f{x)+f<{x)h+'fp^h'+f^li'+--'  +  a,h'^   .    (C) 


EXAMPLE. 
Find  the  result  of  substituting  x  +  h  for  x  in  the  polynomial 
5  x"  -  G  .x-2  -f  8  .T  +  4. 

Here  f(x)  =  5x'-6x''  +  Sx  +  A, 

f'(x)  =  15  x"  - 12  .r  4-  8, 

/"(a;)  =  30.^-12, 

/"'(x)  =  30, 
and  the  result  is 

5.x.3_Ga;2+8a-  +  4  +  (15a;2_i2a;+8)/i  +  (30a;-12)^^-f30.p^. 

81.  Continuity  of  a  Rational  Integral  Function.  Theorem. 
If  in  a  rationed  and  integnd  function  f{x)  the  value  ofx  be  made 
to  vary,  by  indefinitely  small  increments,  from  any  quantity  a  to 
a  greater  quantity  b,  then  loill  f{x)  at  the  same  time  vary  also  by 
indefinitely  small  increments;  that  is,  f(x)  varies  continuously 
ivith  X.  . 


Art.  82  PROPERTIES   OF  POLYNO^fIALS.  103 

Suppose  X  to  increase  from  a  to  a  +  h.  The  correspouding 
increment  of  f{x)  is 

/(«  +  /')  -/(«), 

and,  by  Art.  80,  this  is  equal  to 

/'(a)/i+/"(a)^+...+ca", 

in  which  expression  all  the  coefficients /'(a), /"(a),  etc.,  are 
finite  quantities. 

Now,  by  Art.  79,  Cor.  I,  this  latter  expression  may,  by  tak- 
ing h  small  enough,  be  made  to  assume  a  value  less  than  any 
assigned  quantity ;  so  that  the  difference  between /(a  +  h)  and 
/(a)  may  be  made  as  small  as  we  please,  and  will  ultimately 
vanish  with  h.  The  same  is  evidently  true  during  all  stages 
of  the  variation  of  x  from  a  to  6;  thus  the  theorem  is  proved. 

We  should,  observe  that  it  is  not  here  proved  that  f(x)  in- 
creases continuously  from  f(a)  to  f(b),  but  simply  that  it  varies 
continuously,  for  it  may  sometimes  increase  and  at  other  times 
decrease. 

82.  Form  of  the  quotient  and  remainder  tchen  a  polynomicd 
is  dicided  by  a  binomial. 

Divide  c(qX"  +  a^x"'^  +  a-pf'^  +  •••  4-  f'n-i-i^  4-  «„ 

by  ic  —  h,  and  let  the  quotient  be 

6o.x"-'  +  b^x"-^  +  M""''  +  •••  +  b„_.p:  +  6„_i. 

This  we  shall  represent  by  Q,  and  the  remainder  by  R.  We 
have  then 

f(x)  =  {x-h)Q-\-R. 

The  meaning  of  this  equation  is,  that  when  Q  is  multiplied 
hj  X  —  h,  and.  R  added,  the  result  must  be  identiad,  term  for 
term,  with./'(;c)- 

The  rigM  hand  side  of  the  identity  is 


104 


THEORY   OF  EQUATIONS. 


Art.  8-2 


+     h. 


+  R 


Equating  the  coefficients  of  x  on  both  sides,  we  get  the  fol- 
lowing series  of  equations  to  determine  h^,  bi,  b.,,  •••  &„_i,  B : 

h  =  Clo, 
hi  =  boh  +  ttj, 
62  =bih-j-  Clo, 
bs  =  bih  +  ttg, 


bn-l  =  bn_2ll  -\-  Cln_i, 

R  =  h„_-Ji  +  a„. 


These  equations  supply  a  ready  method  of  calculating  in 
succession  the  coefficients  h^,  b^,  bo,  etc.,  of  the  quotient,  and  the 
remainder  R.  For  this  purpose  we  write  the  series  of  opera- 
tions in  the  following  manner : 


tt'jj  Cfgj  ^25      •*'  ^'7i,_i)         ^^'n? 

boh,      bih,     b.Ji,  •••  hn_Ji     b„_ih, 


bi        &2 


R 


In  the  first  line  are  written  down  the  successive  coefficients 
of  f(x).  The  first  term  in  the  second  line  is  obtained  by  mul- 
tiplying tto  (or  605  which  is  equal  to  it)  by  h.  The  product  bji 
is  placed  under  a^,  and  then  added  to  it  in  order  to  obtain  the 
term  6,  in  the  third  line.  This  term,  thus  obtained,  is  multi- 
plied in  its  turn  by  h,  and  placed  under  a,.  The  product  is 
added  to  a.,  to  obtain  the  second  figure  &2  in  the  third  line. 
Tlie  repetition  of  this  process  furnishes  in  succession  all  the 
coefficients  of  the  quotient,  the  last  figure  thus  obtained  being 
the  remainder.  Tliis  process,  called  Horner's  Method  of  Syn- 
thetic Division,  will  be  made  plain  by  a  few  examples. 


Art.  82  PROPERTIES   OF  POLYNOMIALS.  10; 


The  theorem  of  this  article  is  known  as  the  "  Remaimler 
Theorem." 

EXAMPLES. 

1.   Find  the  quotient  and  remainder  wlien 

2x'  +  4  x"  -  X-  -  16  X  -  12  is  divided  by  a;  +  4. 

Write  the  coefficients  with  —  4  at  their  right  and  proceed  as 
below 


2      4    -1-16    -12  [ 
-  8      16-60      304 

-4 

2  -  4  +  15  -  76  +  292 

Thus  the  quotient  is  2  or'  —  4  x-^  +  15  x  —  76,  and  the  remainder 
is  292. 

2.  Find  Q  and  E  when  3  a;*  -  27  x^  +  14  x  +  120  is  divided 
hyx~G. 

When  any  term  in  a  polynomial  is  absent,  care  must  be  taken 
to  supply  the  place  of  its  coefficient  by  zero  in  writing  down 
the  coefficients  oif(x).  In  this  example,  therefore,  the  calcula- 
tion is  as  follows : 

3        0-27        14        120  [6 
18    108      486      3000 


3  +  18  +  81  +  500  +  3120 
Hence  Q  =  3x^  +  lSx^  +  81  x  +  500,  and  E  =  3120. 

3.  Divide  x*  -  4  x"  -  8  .i-  +  32  by  x-  4. 

1_4    0-8      32  [4 

4    0      0  -  32 
i      0    0-8        0 

In  this  case,  therefore,  Q  =  x^  —  8  and  7?  =  0,  or  the  division 
is  exact,  and  4  is  a  root  of  the  equation  f{x)  =  0. 

4.  Find    Q    and    E,    when    ar' -  4  a;*  +  7aT^- 11  a;  -  13    is 

divided  by  x  —  5. 

Aas.  Q  =  x'  +  .r'  +  12  .i-^  +  60  .c  +  289 ;  E  =  14:52. 


106  THEORY  OF  EQUATIoy^S.  Art.  82 

5.  Find  Q  and  E  Avlieu  x^  +  3  a;'  —  15  x^  +  2  is  divided  by 
x-2. 

6.  Find  Q  and  li  wlien  .x-' +  o.-^  —  10.«  +  113  is  divided  by 
a; +  4. 

83.  Tabulation  of  Functions.  Horner's  synthetic  nietliod  of 
division  aifords  a  convenient  practical  method  of  calculating 
the  numerical  value  of  a  polynomial,  with  numerical  coefficients, 
when  any  number  is  substituted  for  x. 

For,  since 

f{x)  =  {x-h)Q  +  R 

is  an  identical  equation,  it  is  satisfied  by  any  value  whatever 
of  X. 

Let  X  =  h,  then  f(Ji)  =  R,  x  —  h  being  equal  to  zero,  and  Q 
remaining  finite.  Hence  the  result  of  substituting  h  for  x  in 
fix)  is  the  remainder  when  f(x)  is  divided  by  x  —  h,  and  can 
be  calculated  rapidly  by  the  method  of  the  preceding  article. 

For  example,  the  result  of  substituting  —  4  for  x  in  the 
polynomial  of  Ex.  1,  Art.  82,  viz., 

2x^  +  4:  x"  -  ar  -  16  x  -  12, 

is  292,  this  being  the  remainder  after  division  by  x  +  4.  This 
can  be  verified  by  actual  substitution. 

Again,  the  result  of  substituting  5  for  x  in 

x^-4x*  +  7  a;3  -  11  x  -  13 

is  1432,  as  appears  from  Ex.  4,  Art.  82. 

We  saw  in  Art.  81  that  as  x  receives  a  continuous  series  of 
values  increasing  from  —  co  to  +  oc,  f(x)  Avill  pass  through  a 
corresponding  continuous  series.  If  we  substitute  in  succession 
for  X,  in  a  polynomial  whose  coefficients  are  given  numbers, 
a  series  of  numbers  such  as 

5,  -  4,  -  3,  -2,-1,  0,  1,  2,  3,  4,  5,  •••, 


Art.  81 


PROPERTIES   OF  POLYNOMIALS. 


101 


and  calculate  the  corresponding  values  of  f{x),  the  process  may 
be  called  the  tabulation  of  the  function. 


EXAMPLES. 


1.   Tabulate   the  trinomial    2.r  + 
values  of  x: 

_  4,  _  3,  _  2,  -  1,  0,  1,  2,  3,  4 

Values  of  ic   .     . 
Values  of  f{x)  . 


G    for   the  following 


-4 

-3 

_  2 

-1 

0 

1 

2 

3 

4 

oo 

9 

0 

-5 

-6 

-3 

4 

15 

30 

2.  Tabulate  the  polynomial  a;*  —  4  .if'  —  8  a;  +  32  for  the  same 
values  of  x. 

3.  Tabulate  af'  -  G  or'  +  11  x  -  6. 
Values  of  x    . 
Values  of  f{x) 

84.  Graphic  Representation  of  a  Polynomial.  The  values  of 
f(x)  corresponding  to  the  different  real  values  of  x  may  be  con- 
veniently exhibited  to  the  eye  by  a  graphic  representation 
which  we  shall  now  explain. 

t 


-4 

-3 

—  2 

-1 

0 

1 

2 

3       4 

-210 

-120 

-GO 

-24 

-G 

0 

0 

0    +6 

Let  two  straight  lines  OX,  OY  (T\g.  3)  cut  one  another  at 
right  angles,  and  be  produced  indehnitely  in  both  directions. 


108  THEORY  OF  EQUATIONS.  Art.  81 

These  lines  are  called  tlie  x-axis  and  y-axis  respectively. 

Lines,  such  as  Oxi,  measured  on  the  a>axis,  to  the  right  of 
the  y-a,xis,  are  regarded  as  positive;  and  those,  such  as  OA', 
measured  to  the  left,  as  negative.  Lines  parallel  to  YY',  and 
above  the  ic-axis,  such  as  ^P  or  B'Q',  are  positive;  and  those 
below  XX',  such  as  AS  or  A'P',  are  negative.  The  student  of 
Trigonometry  or  Analytic  Geometry  is  already  acquainted 
Avith  these  conventions. 

Any  arbitrary  length  may  now  be  taken  on  OX  as  unity, 
and  any  number,  positive  or  negative,  will  be  represented  by  a 
line  measured  on  XX'.  Inf(x),  give  to  x  the  value  a  and  let 
OA  =  a  ;  calculate  /(a) ;  from  A  draw  AP  parallel  to  0  Y  to 
represent  /(a)  in  magnitude  on  the  same  scale  as  that  on  which 
OA  represents  a,  and  to  represent  by  its  position  above  or 
below  the  line  XX'  the  sign  of  /(a).  OB  =  b,  and  BQ  =f{b), 
would  determine  another  point  Q.  Thus,  corresponding  to  the 
different  values  of  x  represented  by  OA,  OB,  OC,  etc.,  we  shall 
have  a  series  of  points  P,  Q,  R,  etc.,  which,  when  we  suppose 
the  series  of  values  of  x  indefinitely  increased  so  as  to  include 
all  numbers  between  —  co  and  +  oo,  will  trace  out  a  continuous 
curved  line.  This  curve  will,  by  the  distances  of  its  several 
points  from  the  line  OX,  exhibit  to  the  eye  the  several  values 
of  the  function /(aj). 

The  process  here  explained  is  also  called  tracing  the  function 
fix),  and  the  curve  itself  is  often  called  the  cjraph  of  the 
function. 

In  the  practical  application  of  this  method  it  is  well  to  begin 
by  laying  down  the  points  on  the  curve  corresponding  to  cer- 
tain small  integral  values  of  x,  positive  and  negative.  A  curve 
drawn  through  these  points  will  give  at  least  a  general  idea  of 
the  character  of  the  function.  If  we  wish,  at  any  particular 
locality,  to  examine  the  curve  more  minutely,  we  must  take 
several  intermediate  fractional  values  of  x,  and,  of  course,  the 
closer  together  such  points  are  taken,  the  more  accurately  will 
the  function  be  delineated. 


Art.  84 


PROPERTIES   OF  POLiWOMIALS. 


109 


EXAMPLES. 

1.    Trace  the  trinomial  —  or  —  2  j;  +  4  ;  that  is,  find  its  ffmjJi. 
The  unit  of  length  taken  is  one-fourth  of  the  line  OE  in 
Fig.  4. 


The  values  of  f(x)  corresponding  to  integral  values  of  x, 
within  the  limits  of  the  figure,  are  as  follows : 


Values  of  x, 
Values  of /(a-). 


4  I  -3 
4     +1 


+  4 


-1 

+  5 


0 

+  4 


+  1 
+  1 


+  2 
-4 


By  means  of  these  values  we  obtain  the  positions  of  seven 
points  on  the  curve,  A,  B,  C,  D,  E,  F,  G.  This  done,  we  draw 
as  smooth  a  curve  as  we  can  througli  these  points,  whicli  curve 
is  the  required  yraph. 


110 


THEORY  OF  EQUATIONS. 


Art.  84 


2.   Trace  the  polynomial 

10x^  —  nx-  +  x  +  6. 

Tabulating  the  pol^'nomial,  we  have 

Values  of  x,             —3 

_2 

-1 

0 

+  1 

+  2 

+  3 

Values  of /(.i-),    -420 

-144 

—  22 

6 

0 

20 

12G 

We  have  found,  Art.  78,  that  this  function  retains  positive 
values  for  all  positive  values  of  x  greater  than  2.7,  and  nega- 
tive values  for  all  values  of  x  nearer  to  —  co  than  —  2.7. 
The  graph  will,  then,  if  it  cuts  the  axis  of  x  at  all,  cut  it  at  a 
point  (or  points)  corresponding  to  some  value  (or  values)  of  x 
between  —  2.7  and  +  2.7 ;  so,  if  we  wish  simply  to  examine 
the  position  of  the  roots  of  the  equation  f(x)  =  0,  the  tabula- 
tion may  be  confined  to  the  interval  between  —  2.7  and  +  2.7. 

This  is  a  case  in  which  the  substitution  of  integral  values 
only  of  X  gives  little  help  toward  the  tracing  of  the  curve,  and 
where,  consequently,  smaller  intervals  have  to  be  examined.  It 
would  be  well  to  tabulate  the  function  for  intervals  of  one-tenth 
between  the  integers  —  1,  0 ;  0,  1 ;  1,  2,  This  tabulation  and 
the  tracing  of  the  curve  is  left  as  an  exercise  for  the  student. 

3.  Trace  the  trinomial  2  x^  +  x  —  6. 

4.  Trace  the  polynomial  x*  —  15  a;^  +  10  x  +  24. 

The  graph  in  Ex.  1  cuts  the  axis  of  x  in  two  points  (a  num- 
ber equal  to  the  degree  of  the  polynomial) ;  in  other  words, 
there  are  two  values  of  x  for  which  the  value  of  the  given 
polynomial  is  zero;  these  are  the  roots  of  the  equation 
—  a^  —  2a;-l-4  =  0.  It  will  be  found  that  the  graph  of  the 
polynomial  in  Ex.  4  cuts  the  axis  of  x  in  four  points,  corre- 
sponding to  the  roots  of  the  equation 

x'  -  15  .^•2  +  10  .V  +  24  =  0,  viz.  -  4,  -  1,  2,  3. 

The  graph  of  a  given  polynomial  may  not  cut  the  axis  of  x 
at  all,  or  may  cut  it  in  a  number  of  points  less  than  the  degree 


Art.  84  PROPERTIES   OF  POLYNOMIALS.  Ill 

of  the  polynomial.  Such  cases  corresi)uiul  to  the  imaginary 
roots  of  equations,  as  will  appear  more  fully  in  a  subsequent 
chapter.  For  example,  the  graph  of  the  polynomial  2x'+x-\-2 
will  be  found  to  lie  entirely  above  the  axis  of  x.  It  is  evident, 
by  the  solution  of  the  equation  2  jt  +  cc  +  2  =  0,  that  the  two 
values  of  x  which  render  the  polynomial  zero  are  in  this  case 
imaginary.  Whenever  the  number  of  points  in  which  the 
curve  cuts  the  axis  of  x  falls  short  of  the  degree  of  the  poly- 
nomial, it  is  customary  to  speak  of  the  curve  as  cuttinrj  the  line 
in  imaginary  points. 


^^0.^^ 


CHAPTER   VI. 

GENERAL    PROPERTIES   OF   EQUATIONS. 

85.  We  shall  first  prove  some  theorems  which  establish  the 
existence  of  a  real  root  iu  an  equation  in  certain  cases. 

TiiKOKEM.  If  two  real  numbers  substituted  for  x  in  a  rational 
integral  expression  f(x)  give  results  ivith  contrary  signs,  one  root 
at  least  of  the  equation  f(x)  =  0  lies  between  those  values  of  x. 

Let  a  and  b  denote  the  two  numbers ;  then  /(«)  and  /(&) 
have  contrary  signs.  By  Art.  81,  as  x  changes  gradually  from 
a  to  b,  the  expression  f(x)  passes  without  any  interruption  of 
value  from /(a)  to  f(J)) ;  but  since /(a)  and/(&)  are  of  contrary 
signs,  the  value  zero  lies  between  them,  so  that  f(x)  must  be 
equal  to  zero  for  some  value  of  x  between  a  and  b ;  that  is, 
there  is  a  root  of  the  equation  f(x)  =  0  between  a  and  b. 

We  do  not  say  that  there  is  only  one  root;  and  w^e  do  not 
say  that  if  f(a)  and  f(b)  are  of  the  same  sign  there  will  be  no 
root  of  the  equation  f(x')  =  0  betAveen  a  and  b. 

Reference  to  the  graphic  method  of  representation  will  assist 
our  conception  of  this  theorem,  and  will  enable  us  to  make  it 
more  general.  It  is  evident  that  if  there  exist  two  points  of 
the  graph  of  f(x)  on  opposite  sides  of  the  axis  XX\  then  the 
curve  between  these  points  must  cut  that  axis  an  odd  number 
of  times,  and  if  the  two  points  are  on  the  same  side  of  the  axis, 
the  curve  must  cut  that  axis  either  not  at  all  or  an  even  number 
of  times ;  thus  several  values  may  exist  between  a  and  b  for 
which  f{x)  =  0,  that  is,  for  which  the  graph  cuts  the  axis. 

For  example,  in  Ex.  2,  Art.  84,  x  =  —  l  gives  a  negative 
value  (—  22),  and  x  =  +  2  gives  a  positive  value  (20),  and 
112 


Art.  S«      GEN  Eli  AL    rRoPEHTIES   OF  EQUATlnSS.  113 

between  these  points  of  the  curve  there  exist  t/irei'  points  of 
section  with  the  .r-axis,  as  can  be  easily  shown. 

86.  Theorem.  Every  equation  of  an  odd  degree  //a.s  at  le<(nt 
one  real  root  of  a  sign  opposite  to  that  of  its  last  term. 

This  is  evident  at  once  from  the  theorem  of  the  last  article. 
Substitiite  in  succession  —  oc,  0,  oo  for  x  in  the  polynomial 
f{x).     The  results  are,  )i  being  odd  (see  Art.  78), 

for  X  =  —  cc,  f(x)  is  negative ; 

for  X  =  0,  sign  of /(.r)  is  the  same  as  that  of  «„; 

for  X  =  +  oc,  f{x)  is  positive. 

If  a„  is  positive,  the  equation  must  have  a  real  root  between 
—  00  and  0,  i.e.  a  real  negative  root ;  and  if  o„  is  negative,  the 
equation  must  have  a  real  root  between  0  and  cc,  i.e.  a  real 
positive  root.     The  theorem  is  therefore  proved. 

87.  Theorem.  Every  equation  of  an  even  degree,  trhose  last 
term  is  negative,  has  at  least  two  real  roots,  one  jjositive  and  the 
other  negative. 

The  results  of  substituting  —  oo,  0,  oo  are  in  this  case 

-cc,  +,         0,  -,         +^,  +; 

hence  there  is  a  real  root  between  —  oo  and  0,  and  another 
between  0  and  +  <»  ;  i.e.  there  exist  at  least  one  real  negative 
and  one  real  positive  root. 

88.  To  prevent  mistakes,  it  is  well  to  call  attention  to 
exactly  what  has  been  proved  in  the  last  two  articles. 

In  Art.  86  it  is  proved  that  the  equation  considered  has  at 
least  one  real  root:  it  is  not  i)roved  that  it  h:is  only  one.  In 
Art.  87  it  is  proved  that  the  etpiation  considered  lias  at  li-a.^t 
two  real  roots ;  it  is  not  proved  that  it  has  only  two. 


114 


THEORY  OF  EQUATIONS. 


Art. 


89.  Existence  of  a  Root.  Imaginary  Roots.  We  have  now 
proved  the  existence  of  a  real  root  in  the  case  of  every  equa- 
tion, except  one  of  an  even  degree  whose  last  term  is  positive. 

Such  an  equation  may  have  no  real  root  at  all.  We  must 
then  examine  whether  there  may  not  be  cases  where  the  equa- 
tion has  imaginary  roots,  or  whether  there  may  not  be  in  cer- 
tain cases  both  real  and  imaginary  values  of  the  variable 
which  satisfy  the  equation.  In  Chapter  IV  we  have  assumed 
that  such  is  the  case.  Let  us  take  a  simple  example  by  way 
of  illustration. 

In  Art.  84  we  have  seen  that  the  graph  of  the  polynomial 

f(x)  ~2x'-\-x  +  2 

lies  entirely  above  the  axis  of  x,  as  in  Fig.  5.     The  equation 
f(^x)  =  0  has  no  real  roots ;  but  it  has  the  two  imaginary  roots 


as  is  evident  by  the  solution  of  the  quadratic 

We  observe,  therefore,  though 
there  are  no  real  roots,  there  are 
in  this  case  two  imaginary  expres- 
sions which  reduce  the  polynomial 
to  zero. 

The  corresponding  general  propo- 
sition is  that  every  rational  integral 
equation  has  a  root,  real  or  iniagi- 
iiary.  Such  a  root  has  the  general 
form 

a  and  /3  being  real  finite  quantities. 
This  form  includes  both  real  and 
imaginary  roots,  the  former  corre- 
sponding to  the  value  /3  =  0. 


Figr.  5. 


Art  90      GENERAL   PROPEllTIES   OF  EqUATIoSS.  Ho 

The  proof  of  this  fundamental  theorem,  involving  principles 
too  intricate  to  be  introduced  in  an  elementary  treatise,  will 
not  be  given,  and  we  shall  simply  assume  the  i)ropo.siti()ti 
as  true,  referring  the  student  for  the  proof  to  Hurnside 
and  Pan  ton's  Theory  of  Equations,  or  Serret's  Cours  il'AJtjNne 
Siiperieure,  or  any  advanced  work  on  the  subject.* 

90.  Every  Equation  of  the  nth  Degree  has  n  Roots  and  No 
More. 

It  is  evident  from  Art.  ^o  that  if  any  nunibor  h  is  a  root  of 
the  equation  f{x)  =  0,  then  f{x)  is  divisible  by  x  —  h  without 
a  remainder ;  for  if  /(/t)  =  0,  i.e.  if  h  is  a  root  of  /(x)  =  0, 
K  must  =  0. 

Let  the  given  equation  be 

/(.r)  =  .x>"  + /JiX"-'  +;ur"--  +  •••  +p„.iX+p„  =  0. 

This  equation  must  have  a  root,  real  or  imaginary  (Art.  80), 
which  we  shall  denote  by  «i.  Let  the  quotient,  when  f(x)  is 
divided  by  x—a^,  be  <^i(.i');  we  have  then  tlie  identical  ecpiation 

/(.r)  =  (.c-«,)<^,(.r). 

Again,  the  equation  <^,(.i')  =  0,  which  is  of  the  (?i  — l)th 
degree,  must  have  a  root,  which  we  represent  by  rc^.  Let  the 
quotient  obtained  by  dividing  (f)i(x)  hy  x—  a^  be  <^2(-''*)-     Hence 

<^,(.r)  =  (x  -  «,,)  <^.(.r), 

and  .-.  fix)  =  (x  -  «,)  (x  -  «.,)  <t>.^v), 

where  <f>2{x)  is  of  the  (u  —  2)  th  degree. 

Proceeding  in  this  way,  we  prove  that  /(.r)  consists  of  the 
product  of  n  factors,  each  containing  x  in  the  tirst  degree,  and 
a  numerical  factor  <^„(a*)- 

If,  in  the  identical  equation 

f(x)  =  (:x-  «,)  (x  -  a,)  •••  (x  -  «„)  <t>„i.v), 

•  See  also  Fine's  XnmberSijHtfm  <>/  Aijehni,  Artt.  .V.>-,'4. 


116  THEORY   OF  EQUATIONS.  Art.  90 

we  compare  the  coefficients  of  a*",  it  is  plain  that  <^„(.t)  =  1. 
Thus  we  prove  the  identical  equation 

fix)  =  (x  -  «,)  (x  —  «2)  (x  —  «3)  •••  (x  -  «„_i)  (x  -  a„). 

It  is  evident  that  the  substitution  of  any  one  of  the  numbers 
«!,  «2  •••  «„  for  X  in  the  right-hand  member  of  this  equation 
will  reduce  that  member  to  zero,  and  will,  consequently,  reduce 
f(x)  to  zero ;  that  is,  the  equation  f(x)  —  0  has  for  roots  the 
n  quantities  «j,  «2,  «3  •••  a„_i,  «„.  And  it  can  have  no  other 
roots ;  for  if  any  number  other  than  one  of  the  numbers  «i, 
«2,  «3  •••  «„  be  substituted  in  the  right-hand  member  of  the 
above  equation,  the  factors  will  all  be  different  from  zero, 
and,  therefore,  the  product  cannot  vanish. 

This  theorem,  while  of  no  assistance  in  the  solution  of  the 
equation  f(x)  =  0,  enables  us  to  solve  the  converse  problem ; 
that  is,  to  find  the  equation  whose  roots  are  any  n  given 
quantities.  The  required  equation  is  obtained  by  multiplying 
together  the  n  simple  factors  formed  by  subtracting  from  x 
each  of  the  given  roots. 

It  follows  also  from  the  present  theorem  that,  when  any 
(one  or  more)  of  the  roots  of  a  given  equation  are  known,  we 
can  obtain  the  equation  containing  the  remaining  roots  by 
dividing  the  given  equation  by  the  given  binomial  factor  or 
factors.  The  quotient  will  be  the  required  polynomial  com- 
posed of  the  remaining  factors. 

EXAMPLES. 

1.  Find  the  equation  whose  roots  are 

2,   -1,  -4,   -f  3.     Ans.  x* -15x' +  10x  +  24:  =  0. 

2.  Two  of  the  roots  of  the  equation 

X*  -  5  x^  -ISx""  +  53x  +  m  =  0 

are  —3,  +4;  find  the  other  roots.     Use  the  method  of  division 
of  Art.  82. 


Art.  91      GENERAL   PROPERTIES   OF  EQirATlONS.  117 

3.  Find  the  eciuatioii  whose  roots  are 

-2,     0,     +1,     +5. 

4.  In  the  equation 

x'-Sa^-lGx  +  iS, 
one  root  is  —  4  ;  find  the  other  roots. 

5.  Solve  the  equation 

X*  -  IG  x^  +  86  .r'  -  176  x  +  105  =  0, 
two  roots  being  1  and  7.  Ans.  Other  roots  8,  o. 

6.  Form  the  equation  whose  roots  are 

-|,  2,   +4.     Ans.  16x^  +  37x^  +  12x--i  =  0. 

7.  Solve  the  equation 

^.4_4a-3_8.f  +  32  =  0, 
two  roots  being  —  1  +  V—  3,  —  1  —  V—  3. 

8.  Solve  the  cubic  equation 

x'  -1  =  0. 

Here  it  is  evident  that  x  =  1  satisfies  the  equation.  Divide 
by  X  —  1,  and  solve  the  resulting  quadratic  to  get  the  other 
two  roots. 

9.  Solve  the  cubic  equation 

ar*  +  1  =  0. 

91.  Equal  Roots.  It  is  evident  that  the  n  factors  of  which 
a  })()lyiioniiaI  f(x)  consists  need  not  be  all  different  from  one 
another.  The  factor  x  —  u,  for  example,  may  occur  in  the 
second  or  any  higher  powder  not  superior  to  n.  In  this  ca.so 
tw^o  or  more  of  the  n  roots  of  f(x)  are  equal  to  one  another,  and 
the  root  a  is  called  a  multiple  root  of  the  equation,  —  double, 
triple,  etc.,  according  to  the  number  of  times  the  factor  is 
repeated. 


118  THEORY  OF  EQUATIONS.  Art.  91 

Equal  roots  form  the  connecting  link  between  real  and 
imaginary  roots.  A  reference  to  the  graphic  construction 
(Art.  84)  will  make  this  plain.  Or,  returning  to  the  equation 
given  in  Art.  49,  we  know  that  the  two  roots  of  the  equation 
ax^  +  bx  +  c  =  0,  are  real,  if  ¥  >  4  ac,  equal,  if  6^  =  4  ac,  and 
imaginary,  if  Z>-  <  4  ac. 

92.  Theorem.  Li  an  equation  ivith  real  coefficients,  complex 
roots  occur  in  pairs. 

Let  f(x)  be  a  rational,  integral  function  of  x  in  which  the 
coefficients  are  all  real;  then  if  a  +  ^V— 1  is  a  root  of  the 
equation  f(x)  =  0,  a  —  ftV^^  will  also  be  a  root. 

For  when  a  +  ^V—  1  is  put  for  x,  the  function  f{x)  takes 
the  form  P  +  Q/3^/—l,  where  Pand  Q  involve  even  powers  of 
^.  Now  as  the  coefficients  in  f{x)  are  supposed  real,  V—  1  can- 
not  occur  except  with  some  odd  power  of  /3.  If  then  a— ySV  — 1 
be  substituted  for  x  in  f(x),  the  result  will  be  obtained  by 
changing  the  sign  of  ^  in  the  result  obtained  by  substituting 
«  _(_  /3V^^  for  x;  the  result  is  therefore  P  —  Q^V^^.  (Art. 
61,  Cor.  2.) 

Now  if  a  +  /3V—  1  is  a  root  of  f{x)  =  0,  then 

and,  therefore,  Art.  57,  since  (3  is  not  zero, 
P=0,  and  Q  =  0. 
Hence  P-QJ^  V^l  =  0, 

and  a  —  I3\  —  _  is  also  a  root  of  f(x)  =  0.  Thus  the  total 
number  of  imaginary  roots  in  an  equation  with  real  coefficients 
is  always  even. 

Note.  A  proof  exactly  similar  to  that  above  given  shows  that  surd  roots, 
of  the  form  a  ±  Vy,  enter  in  pairs  equations  ivhose  coefficients  are  rational. 


Art.  93 


GENERAL    PROPERTIES   OF  EQUATIOXS.  110 


EXAM  PLES, 

1.  Form  a  rational  cubic  equation  which  shall  liave  for  two 
of  its  roots 

1,  3-2V^ri. 

2.  Form  a  rational  ecjuation  which  shall  have  for  two  of  its 
roots 

Ans.  X*  -  12  x"  +  72  x'  -  312  x  +  G7G  =  0. 

3.  Solve  the  equation 

X*  -  x^  -  8  .r^  +  8  =  0, 
which  has  a  root  1  +  VS. 

4.  Solve  the  equation 

2^?  —  o(?  —  Q>x-{-  77, 
one  root  being  2  +  V—  7.  Ans.  2  ±  V—  7,  —  |. 

93.  Descartes'  Rule  of  Signs.  This  celebrated  theorem  of 
Descartes*  establishes  an  interesting  and  useful  relation 
between  the  number  of  changes  of  sign  of  the  first  member 
of  an  equation,  f(x)  =  0,  and  the  number  of  real  roots,  and, 
thereby,  enables  us  to  find  a  superior  limit  to  the  number  of 
positive  and  negative  real  roots  of  an  equation. 

Definition.  "When  each  term  of  a  set  of  terms  has  one  of 
the  signs  +  or  —  before  it,  then  in  considering  the  terms  in 
order,  a  contimmtion  is  said  to  occur  when  a  sign  is  the  same 
as  the  immediately  preceding  sign,  and  a  change^  is  said  to 
occur  when  a  sign  is  contrary  to  tlie  immediately  preceding 
sign.     Thus  in  the  expression 

a.-8  _  2 .1^  -  3  .f«  +  4  or'  +  x^  +  2 .1-^  -  3  jr^  -  x  +  1 

*  Rene  Descartes  (l.'>!)f»-ir..50). 

+  Instead  of  "continuation"  and  "change"  the  terms  permanenct  and  rariation  aro 
often  used. 


120  THEORY   OF  EQUATIONS.  Art.  93 

there  are  four  continuations  and  four  changes.  It  is  obvious 
that  in  any  comi)lete  equation  the  number  of  continuations  to- 
gether with  the  number  of  changes  is  equal  to  the  number 
which  expresses  the  degree  of  the  equation.  If  in  any  com- 
plete equation  we  put  -  x  for  x,  the  continuations  and  changes 
in  the  original  equation  become  respectively  changes  and  con- 
tinuations in  the  new  equation. 

(a)   Positive  Roots. 

Theorem.  No  equation  can  have  more  positive  real  roots 
than  it  has  changes  of  sign  from  +  to  —,  and  from  —  to  +, 
in  the  terms  of  its  first  member. 

Let  the  signs  of  a  polynomial  taken  at  random  succeed  each 
other  in  the  following  order : 

-f  +  -  +  --  +  +  +  -  +  -  + 

In  this  there  are  in  all  eight  changes  of  sign.  It  is  proposed 
to  show  that  if  this  polynomial  be  multiplied  by  a  binomial 
whose  signs,  corresponding  to  a  positive  root,  are  -\ — ,  the 
resulting  polynomial  will  have  at  least  one  more  change  of 
sign  than  the  original.  Writing  down  only  the  signs  that 
occur  in  the  operation,  we  have 

+  - 


--  +  -  +  + +  -  + 

+±-+-T+±±-+-+- 


Here,  in  the  result,  the  ambiguous  sign  ±  is  placed  wher- 
ever there  are  two  terms  with  different  signs  to  be  added. 
We  readily  see  that  in  this  case,  and  in  any  other  arrangement, 
the  effect  of  the  process  is  to  introduce  the  ambiguous^  sign 
wherever  the  sign  +  follows  -f ,  or  —  follow^s  — ,  in  tlie  orig- 
inal polynomial.  The  number  of  variations  of  sign  is  never 
diminished,  and  there  is  always  one  variation  added  at  the 


Art.  93      GEXERAL   PliOPKliTIES   OF  EQUATIONS.  llil 

end.  By  trying  different  arrangements  of  signs,  it  is  easy 
to  convince  ourselves  that,  in  even  the  most  unfavorable  case 
—  that,  namely,  in  "svhich  the  continuations  of  sign  in  the 
original  remain  continuatious  in  the  resulting  polynomial, — 
there  is  one  variation  added.  We  may  conclude  in  general 
that  the  effect  of  the  multiplication  of  a  polynomial  by  a 
binomial  x  —  a  is  to  introduce  at  least  one  change  of  sign. 

Now  suppose  we  have  a  polynomial  formed  of  the  product 
of  the  factors  corresponding  to  the  negative  and  imaginary 
roots  of  an  equation.  The  effect  of  multiplying  this  by  each 
of  the  factors  x  —  «,  x  —  (3,  x  —  y,  etc.,  corresponding  to  the 
positive  roots  «,  /3,  y,  etc.,  is  to  introduce  at  least  one  change 
of  sign  for  each ;  so  that  when  the  complete  product  is  formed 
containing  all  the  roots,  we  conclude  that  the  resulting  poly- 
nomial has  not  more  positive  roots  than  there  are  changes  of 
sign. 

(6)  Negative  Roots. 

Theorkm.  Xo  equation  can  have  a  greater  ninnber  of  necja- 
tive  roots  than  there  are  changes  of  sign  in  the  terms  of  the  yolij- 
nomial  f(—x). 

Now,  if  —  X  be  substituted  for  a;  in  the  equation  f(x)  =  0, 
the  resiilting  equation  will  have  the  same  roots  as  the  original, 
except  that  their  signs  will  be  changed;  for,  from  the  identical 
equation 

/(.r)  =  (.f  -  «i)  (:c  -  «.,)  (x  -  «,)  •••  (.r  -  «„), 

we  derive 

/(-  x)  =  {-  1)"  (x  +  «,)  (x  +  «-.)  (x  +  Us)  ■■'(x  +  «„). 

From  this  it  is  evident  that  the  roots  of  /(—  x)  =  0  are 

—  «i,   —  a-,,   —  «3,  •••  —  u„. 

Hence  the  negative  roots  oif(x)  are  positive  roots  of /(— x), 
and  our  theorem  for  negative  roots  is  true. 


122  THEORY   OF  EqUATIONS.  Art.  98 


EXAMPLES. 

1.  If  the  coefficients  in  /(.«)  are  all  positive,  the  equation 
f{x)  =  0  has  no  positive  root. 

2.  If  the  coefficients  in  any  complete  equation  be  alter- 
nately positive  and.  negative,  the  equation  cannot  have  a  nega- 
tive root. 

3.  If  an  equation  consist  of  a  number  of  terms,  whose 
coefficients  are  positive  followed  by  a  number  of  terms  whose 
coefficients  are  negative,  it  has  one  positive  root  and  no  more. 

Apply  Art.  85  and  Art.  93. 

4.  If  an  equation  contain  only  even  powers  of  x,  and  if  all 
the  coefficients  have  positive  signs,  it  cannot  have  a  real  root. 

5.  If  an  equation  contain  only  odd  powers  of  x,  and  if  all 

the  coefficients  have  positive  signs,  it  has  the  root  zero  and  no 
other  real  root. 

6.  Find  an  inferior  limit  to  the  number  of  imaginary  roots 
of  the  equation 

x"  -  3  .^•2  -  a;  +  1  =  0. 

Here,  Art.  93,  the  arrangement  of  signs  for  f(x)  =  0 
+ + 

exhibits  two  changes  of  signs,  hence  there  cannot  be  more 
than  two  positive  roots ;  and,  examining  the  arrangement  for 
/(—  a;)  =  0,  -f-  —  +  -f,  we  find  again  two  changes  of  sign,  so 
there  cannot  be  more  than  two  negative  roots.  As  there  are 
six  roots  in  all,  it  follows  that  there  must  be  at  least  two 
imaginary  roots. 

7.  Find  an  inferior  limit  to  the  n\imber  of  imaginary  roots 
of  the  equation 

x^  +  3x'-{-4x'+2x-6  =  0. 


At  least  four  imaginary  roots. 


Art.  98      GENERAL   PROPEIiTIES   OF  EQUATIONS.  l'2:\ 

8.  Find  the  nature  of  the  roots  of  the  equation 

x'  +  15  x"  +  7  a;  -  11  =  0. 

Ans.  One  positive,  1  negative,  2  imaginary. 

9.  Show  that  the  equation 

x^  +  qx  4-  ?•  =  0, 

where  q  and  r  are  essentially  positive,  has  one  negative  and 
two  imaginary  roots. 

10.  Find  the  nature  of  the  roots  of  the  equation 

x^  —  qx  +  ?•  =  0. 

11.  Show  tliat  the  equation 

X"  -1  =  0 

has,  when  n  is  even,  two  real  roots,  —  1  and  +  1,  and  no  otlier 
real  root ;  and,  when  n  is  odd,  the  real  root  1,  and  no  other 
real  root. 

12.  Show  that  the  equation 

X"  +  1  =  0 

has,  when  n  is  even,  no  real  root ;  and,  when  n  is  odd,  the  real 
root  —  1,  and  no  other  real  root. 


CHAPTER  VII. 

RELATIONS    BETWEEN    ROOTS    AND    COEFFICIENTS.— 
SYMMETRIC   FUNCTIONS. 

94.   Relations  between  the  roots  and  coefficients  of  an  equation. 
Kepresenting  the  n  roots  of  the  equation 

X"  +  Pi.T"-!  +  2hx"-^  -\ 1-  Pn-iX  +Pn    '     '     '      (1) 

by  «i,  «2,  «3,  •••  «„,  Ave  have  the  identity 

Ct-"  +  piX"-^  +  j9oX"--  H h  2)n-lX  +  Pn 

=  (x-a{)(x  —  a.,)(^x-a.;)--'(x  —  a„)    ...     (2) 

When  the  factors  of  the  second  member  of  this  identity  are 
multiplied  together,  the  highest  power  of  x  in  the  product  is 
X",  and  the  coefficient  of  this  term  is  unity.  The  coefficient  of 
the  second  term,  a;""^,  is  —  a^  —  a.^—  a^—  a^---  —  a„;  that  is, 
the  sum  of  the  roots  with  their  signs  changed ;  the  coefficient 
of  x""'^  is  the  sum  of  the  products  of  the  roots  taken  two  and 
two ;  the  coefficient  of  x"'^  is  the  sum  of  the  products  of  the 
roots  taken  three  at  a  time,  with  their  signs  changed ;  and  so 
on,  the  last  term  being  the  product  of  all  the  roots  Avith  their 
signs  changed.  Therefore,  equating  coefficients  of  like  powers 
of  X  on  each  side  of  the  identity  (2),  we  have 


2h  =  -  («i  +  «,  +  «3  +  •••  +  a,,)  ] 
^2  =  («i«2+ «i«.i+ •••)  I 

2h  =  —  (filfij(^  +  «l'<.;«4  +   •••) 

J  24 


(3) 


Art.  95  ROOTS  AND    COEFFICIENTS.  125 

These  results  give  us  the  following  relations  between  the 
roots  and  coefficients : 

In  every  ahjebndc  equation,  the  coefficient  ofichose  highest  term 
is  unity,  the  coefficient  jJi  of  the  second  term,  icith  its  sign  changed, 
is  equal  to  the  sum  of  the  roots. 

The  coefficient  jh  of  the  third  term  is  equal  to  the  sum  of  the 
products  of  the  roots  taken  two  by  two. 

The  coefficient  2h  of  the  fourth  term,  loith  its  sign  changed,  is 
equal  to  the  sum  of  the  jyroducts  of  the  roots  taken  three  by  three, 
and  so  on,  the  signs  of  the  coefficients  being  alternately  negative 
and  positive,  till  finally  that  function  is  reached  which  consists  of 
the  product  of  the  n  roots. 

When  the  coefficient  Uq  of  x"  is  not  unity  (Art.  51),  we  must 
divide  each  term  of  the  equation  by  it. 

Cor.  I.  Every  root  of  an  equation  is  a  divisor,  whole  or 
fractional,  of  the  absolute  term  of  the  equation. 

Cor.  II.  If  the  roots  of  an  equation  be  all  positive,  the 
coefficients  (including  that  of  the  highest  power  of  x)  will  be 
alternately  positive  and  negative ;  and  if  the  roots  be  all 
negative,  the  coefficients  will  be  all  positive. 

95.  It  might  perhaps  be  supposed  that  the  relations  given 
in  the  preceding  article  Avould  enable  us  to  find  by  elimination 
the  roots  of  any  proposed  equation;  for  they  furnish  equa- 
tions involving  the  roots,  and  the  number  of  these  equations 
is  the  same  as  the  number  of  the  roots.  But  this  is  not  the 
case,  for,  on  attempting  this  elimination,  we  merely  reproduce 
the  j)roposed  equation  itself,  as  the  following  example  will  show  : 

Let  «,  p,  y  be  the  roots  of  the  cubic  equation 

ar^+iV^''  +  i>L'^-  +  ^'3  =  0 0) 

We  have,  by  Art.  94, 

p,^-(a  +  f3  +  y), 
p,  =  (c(3  +  ay  +  (3y. 
Pi  =  -  «^y- 


126  THEORY  OF  EQUATIONS.  Art.  95 

Multiplying  the  ftrst  of  these  equations  by  a^,  the  second 
by  a,  and  adding  the  three,  we  find 

or  rr  +  jh"-'  +  i^2«  +  Ih  =  0> 

which  is  the  given  cubic  with  a  in  the  place  of  x,  and,  there- 
fore, we  are  no  nearer  the  solution  of  (1)  than  v/e  Avere  at  first. 
Thus,  although  the  equations  (3)  afford  no  aid  in  the  general 
solution  of  the  equation,  they  are  often  useful  in  facilitating 
the  solution  of  nun^erical  equations  when  any  particular  rela- 
tions among  the  roots  are  known  to  exist,  as  will  be  made 
apparent  by  the  following  examples. 

EXAMPLES, 

1.  Solve  the  equation 

two  of  its  roots  being  equal. 

Let  a,  a,  /3  be  the  three  roots.     We  have 
2a  +  (3  =  3, 
«2  +  2  «/3  =  0, 
from  which  we  find  a  =  2,  |8  =  —  1.     The  roots  are  2,  2,  —  1. 

2.  Solve  the  equation 

a,-3_  5.^2  _  16a; +  80  =  0, 
the  sum  of  two  of  its  roots  being  zero. 
Let  the  roots  be  «,  jS,  y.     We  have  then 
«  +  /8  +  7=    5, 
a{3  +  ay  +  (3y  =  -  16, 

(i/3y  =  -  80. 

Taking  ^  +  y  =  0,  we  get  «  =  5,  /S  =  4,  y  =  -  4.     Thus  the 
three  roots  are  5,  4,  —  4. 


Art.  95  ROOTS  AND   COEFFICIENTS.  V21 

3.  The  equation 

x'  _  4  x"  -  12  .r-'  +  32  a;  +  64  =  0 
has  two  pairs  of  equal  roots ;  find  them. 

4.  Solve  the  equation 

x'-9x'  +  Ux  + 2-1  =  0, 
two  of  whose  roots  are  in  the  ratio  of  3  to  2. 

Let  the  roots  be  a,  /?,  y,  with  the  relation  2a  =  2  ^. 

Ans.  The  roots  are  6,  4,  —  1. 

5.  Solve  the  equation 

X*  -f  2  ar'  -  21  .x-2  -  22  a;  4-  40  =  0, 
whose  roots  are  in  arithmetical  progression.     Assume  for  the 
roots  a  — 3  8,  a  —  8,  a  +  8,  a  +  3  8. 

6.  Solve  the  equation 

8  .K-*  -  30  •r'  +  35a^  -  15  x  +  2  =  0, 

whose  roots  are  in  geometrical  progression.     Assume  for  the 

roots  -,  -,  up,  ap\  Ans.  \,  \,  1,  2. 

P    P 

7.  Solve  the  equation 

x^-3x^-x  +  3  =  0, 
whose  roots  are  in  arithmetical  progression. 

8.  Solve  the  equation 

24.r^-2Grr  +  9.r-l  =  0, 
whose  roots  are  in  harmonic  progression.  Ans.  ^,  J,  }. 

9.  Solve  the  equation 

X*  +  15  x"  +  70  ar  +  120  x  +  CA  =  0, 
whose  roots  are  in  geometric  progression. 

10.  The  equation 

3  X*  -  25  ar'  +  50  x^  _  r,o  x  +  12  =  0 
has  two  roots  whose  product  is  2 ;  find  all  the  roots. 


128  THEORY    OF    EQUATIONS.  Art.  9G 

96.  Derived  Functions.  In  order  to  examine  an  equation  for 
equal  roots,  it  will  be  found  convenient  to  express  the  derived 
functions  (Art.  80)  in  another  form. 

Let  the  roots  of  the  equation  f(x)  =  0  be  «i,  «2)  «35  •  •  •  ««• 
We  have 

/(:^•)  =  (.«  -  «i)  (x  -  «2)  (x  -  «3)  ■•■(x-  «„). 

In  this  identical  equation  substitute  h  +  x  for  x: 

■X  (h  +  x)  =  (h  -\-  X  —  «i)  (h  +  x  —  a^  •  •  •  (h  -\- x  —  «„) 

~"  =  A"  +  gi/i"-i  +  g2^t''-2  -I \-  q„_ih  +  g,„ 

where 

gi  =  ;^-  —  «i  +  a;  —  «2  +  -t"  —  «3  + h  S''  —  «„, 

g,  =  (x  -  «i)  (.^•  -  «2)  +  (-c  -  «i)  (x-  -  «3)  H \-(x  —  «„_,)  (.r  -  «„), 

g„_i  =  (.^•  —  «,)  (x  —  fta)  •••(.«  —  «„)  +  (ic  —  «i)  (a;  —  «3)  •  •  •  (.r  —  «„)  +" 

• .  •  +  (a;  -  tti)  (.T  -  «2)  •  •  •  (it-  -  «„_i), 
g„  =  (x  —  ai)(x  —  a^)  (x  —  Wg)  •••  (x  —  «„). 
Also  we  have,  by  Art.  80, 

f(h  +  a-)  =/(.r)  +./"(.i-)7i  +-y~^/r  +  -  +  /i". 

Equating  the  two  expressions  for  f(h  +  a:),  we  obtain 
f(x)  =  (w  -  «i)  (x  —  «2)  •  •  •  (a;  —  «„), 

f'(x)  =  (x  —  a.,)  (x  —  (is)  •••  (x  —  «„)  +  •••,  as  above  written, 

f"(x) 

^ — rr  —  tlie  similar  value  of  g„_2  in  terms  of  a;  and  the  roots. 

The  value  of/'(.c)  may  be  conveniently  written  as  follows: 

n.)=i<±+i(-'L+...+i(^.. 

X  —  Ui      X  —  a.2  x  —  a„ 


0^ 


Alt.  98  ROOTS  AND   CO  EFFICIENTS.  \-l\) 

97.   Multiple  Roots. 

TiiEouKM.  .1  multiple  root  of  the  order  p  of  the  etjiiation 
f{x)  =  0  is  a  multiple  root  of  the  order  p  —  1  of  the  first  dirired 
equation  f  (x)  =  0. 

This  follows  at  once  from  the  expression  given  for  f  (x)  in 
the  i)receding  article;  for,  if  the  factor  {x  —  u^Y  occnrs  in 
fix),  that  is,  if  «i  =  Uo  =  •  •  •  =  «^,,  we  have 


Each  term  of  this  will  still  have  {x  —  «,  p  as  a  factor,  ex- 
cept the  first,  Avhich  will  have  (.r  —  «,)''-'  as  a  factor ;  lience 
{x  —  «i)^~^  is  a  factor  in/'(a'). 

Cor.  I.  Any  root  which  occnrs  p  times  in  the  eqnation 
f(x)  =  0  occurs  in  degrees  of  multiplicity  diminishing  by 
unity  in  the  first  p  —  1  derived  equations. 

Since /"(.t;)  is  derived  from  f'(x)  in  the  same  manner  as 
f'{x)  is  from  f(x),  it  is  evident  by  the  above  theorem  that 
f"{x)  Avill  contain  {x  —  a^Y'-  as  a  factor.  Tlie  next  derived 
function, /'"(ic),  will  contain  (x  —  UiY'^;  and  so  on. 

98.   Determination  of  Multiple  Roots. 

From  the  preceding  article  it  is  obvious  that  if  f(x)  antl 
f'(x)  have  a  common  factor  (x  —  «/"',  (x  —  «V  will  be  a  factor 
inf{x);  hence  a  is  a  root  of  f(x)  of  multiplicity  p.  In  the 
same  way,  it  appears  that  if  f(x)  and/'(j:)  liave  other  com- 
mon factors  (x  —  /3)'~^  (x  —  y)*""',  {x  —  8)'  ',  etc.,  the  efpiation 
/(.c)  =.  0  will  have  q  roots  equal  to  /?,  r  roots  equal  to  y,  .s  i-oots 
equal  to  8,  etc. 

Hence,  in  order  to  examine  an  equation /"(.r)  for  e(|ual  roots 
and  to  determine  these  roots,  if  such  exist,  we  must  find  the 
highest  common  factor  of /(.r)  and./*'(.r).  Let  this  ho  F,(.r)=0. 
The  solution  of  Fi{x)  —  0  will  give  the  equal  roots. 


130  THEOBY  OF  EQUATIONS.  Art.  98 

EXAMPLES. 
1.   Find  the  multiple  roots  of  tlie  equation 
x"  -  7  .r  +  16  X  -  12  =  0. 

Here  the  H.  C.  Y.  of  f(.v)  =  or'  -  7  x-  +  16x-  12,  and  /'  (.r) 
=  3  X-  -  14  X  +  16  is  .i;  -  2  ;  hence  {x  -  2)^  is  a  factor  in  /(.«). 

The  other  factor  is  x  —  3,  hence  the  roots  of  the  equation 
are  2,  2,  3. 

Whenever,  after  determining  the  multiple  factors  of  f(x), 
we  wish  to  get  the  remaining  factors,  it  will  be  convenient  to 
apply  Horner's  method  of  division  (Art.  82).  In  this  example 
we  would  divide  twice  by  ^  -  2,  the  calculation  being  repre- 
sented as  follows 

1     -7     +16     -12 
2     - 10     +12 

1-5  6  0 

9         _  A 


1-3  0 

Thus  1  and  —  3  being  the  two  coefficients  left,  the  third  fac- 
tor is  X  —  3.  This  operation  verifies  the  previous  result,  the 
remainders  after  each  division  vanishing  as  they  ought. 

2.  Find  the  multiple  roots,  and  the  remaining  factor  of 
the  equation 

a,-5  _  10  :v-  +  15  a;  -  6  =  0. 

The  H.  C.  F.  of  f(x)  and  /'  (x)  is  x''-2  x  +  1.  Hence  (;x-lf 
is  a  factor  in  f(x).  Dividing  three  times  in  succession  by 
x  —  1,  we  obtain 

f(x)  =  (x-iy(x'  +  3x-\-6). 

Find  the  multiple  roots  of  the  following  equations : 

3.  o!^  +  x'-16x  +  20  =  0. 

4.  x*-2  x"  -  11  a;^  +  12  x  +  36  =  0. 


Art.  99  ROOTS  Ay D   VOEFFICIENTS.  1:}1 

5.  x^-llar  +  18;«-8  =  0. 

6.  X*  -  11  af*  +  44  X-  -  76  x  +  48  =  0. 

A„s.  f(x)  =  (x  -  2y(x  -3){x-  4). 

7.  2x*-12x'  +  ldx--ex  +  9  =  0. 

Ans.  The  roots  are  3,  3,  +  jV—  -,  —  .VV—  2. 

8.  Show  that  the  binomial  equation 

x"  —p"  —  0 
cannot  have  equal  roots. 

9.  Apply  the  method  of  Art.  08  to  determine  the  condition 
that  the  cubic 

.r'  -f  3  ZTx-  +  (^  =  0 

should  have  a  pair  of  equal  roots.  Ans.  C^  +  4  H^  =  0. 

The  ordinary  process  of  finding  the  H.  C.  F.  of /(a*)  and/'(x') 
may  often  become  very  laborious.  It  is  chiefly  in  connection 
with  Sturm's  theorem  (Art.  118)  that  the  operation  is  of  any 
practical  value.  Multiple  roots  of  equations  of  degrees  inferior 
to  the  sixth  can  be  determined  more  readily  by  trial. 

99.  Theorem.  —  In  jyassing  continnonsJ}/  from  a  value  a  —  // 
of  X  a  little  less  than  a  real  root  a  of  the  erpiation  f(x)  :=0  to  a 
valne  a  +  h  a  little  greater,  the  pob/iwmials  f(x)  and  f'(x)  hare 
unlilce  sights  immediately  before  the  jmssage  through  the  roof,  and 
like  signs  immediateli/  after. 

Substituting  a  —  h  in /(.r)  and  ,/''(•'•)>  ^"<^  expanding,  we  have 
/(«  -  h)  =/(«)  -f'(u)h  +-^~'4fr  -  ..., 
f'(a-h)=  f\u)    -f"(u)h    +-. 

Now,  since  ./(«)  =  ^i  the  signs  of  these  expressions,  dejiend 
ing  on  those  of  their  first  terms,  are  unlike.  Wliou  the  .sign 
of  h  is  changed,  tlie  signs  of  /(«<  +  h)  and  /'(«  +  h)  are  like. 
Hence  the  theorem. 


132  THEORY  UF  EQUATIONS.  Art.  100 

100.   The  Cube  Roots  of  Unity.     Equations  of  the  forms 

x"  —  p  =  0,  .t"  -|-  j;  =  0, 

are  called  hinomial  equations.  We  shall  see  later  that  such 
equations  are  intimately  connected  with  the  more  special  forms 

a;"  -  1  =  0,  X"  +  1  =  0, 

the  roots  of  the  first  of  which  are  called  the  n  nth  roots  of 
unity.  We  shall  here  consider  the  simple  case  of  the  binomial 
cubic. 

We  have  seen  (Ex.  1,  Art.  73)  that  the  roots  of  the  cubic 

ar'  -1  =  0 

are  1,  -i  +  |V^^,  -  i  -  i V^=^. 

(See  also  Ex.  8,  Art.  90.) 

If  either  of  the  imaginary  roots  be  represented  by  w,  the 
other  is  easily  seen  to  be  <o^  by  actual  squaring.  We  have 
then  the  identity 

.^'^  —  1  =  {x  —  1)  (x  —  w)  {x  —  o)-). 

Changing  x  into  —  x,  Ave  get  the  following  identity  also  : 

x"  +  1  =  (.-c  +  1)  (.^•  +  0.)  (x  +  W-), 

which  gives  the  roots  of     x^  4-1  =  0. 

Whenever  cu  raised  to  any  higlier  power  than  the  second 
presents  itself,  it  can  be  replaced  by  w,  or  wr,  or  1 ;  for  example 

oi*  =  w'^   •   W  ^=  w,    w^  =  cu'^   •   W^  =  0)^, 

w''  =  0)^  •  w''  =  1,  etc. 

By  the  first  or  second  of  equations  (3),  of  Art.  94,  we  have 
the  following  property : 

1  +  O)  +  CO-  =  0. 


Alt.  101  BOOTS  A^'D   COEFFICIEyTS.  IS'6 

Cor.     It  is  important  to  observe  that,  corresponding  to  the 
n  »th  roots  of  unity,  there  are  n  nth  roots  of  any  quantity. 
The  roots  of  the  equation 

x"  —  a  =  0 
are  the  n  nth.  roots  of  a. 

The  three  cube  roots,  for  example,  of  a  are 
Va,  wA,/a,  w^Va, 

where  V«  represents  the  ordinary  (real)  cube  root.     Each  of 
these  values  satisfies  the  cubic  equation  or'  —  a  =  0. 

Thus,  besides  the  ordinary  cube  root  3,  the  number  27  has 
the  two  imaginary  cube  roots 

3^    I    3.-y/ 3     3. 3  -y/  _  3 

as  the  student  can  easily  verify  by  actual  cubing. 

EXAMPLES. 

1.  Show  that  the  product 

(wm  +  o)-n)  (wrni  +  wn) 
is  rational.  Ans.  m?  —  mn  -\-  n^. 

2.  Show  that  the  product 

(«  +  w^  +  a>7)  («  +  W-/3  +  wy) 
is  rational. 

3.  Form  the  equation  whose  roots  are  m  +  ?j,  mm  4-  <i>-h, 
(ji'ni  -\-  0)71. 

101.  Symmetric  Functions  of  the  Roots.  Symmetric  func- 
tions of  the  roots  of  an  equation  are  those  which  are  not 
altered  if  any  two  of  the  roots  be  interchanged.  For  example, 
if  «,  /8,  y  are  the  roots  of  a  cubic  equation,  « +  ^  +  y, 
«)8  +  «y  +  j8y,  afiy  are  symmetric  functions,  for  all  the  roots 
are  involved  alike.     The  functions  p„  p^,  2^3,  etc.,  of  Equation  3, 


134  THEOBY.   OF  EQUATIONS.  Art.  101 

Art.  94,  are  the  simplest  symmetric  functions  of  the  roots,  each 
root  entering  in  the  first  degree  only  in  any  one  of  them.  We 
can  often,  as  shown  by  some  examples  appended  to  this  article, 
obtain  the  values  of  a  great  variety  of  symmetric  functions  in 
terms  of  the  coefficients  of  the  equation  whose  roots  we  are 
considering. 

A  symmetric  function  is  usually  represented  by  the  Greek 
letter  2  attached  to  one  term  of  it,  from  which,  by  analogy,  the 
entire  expression  may  be  written  down. 

Thus,  in  the  case  of  a  cubic,  whose  roots  are  «,  (3,  y, 

where  all  possible  products  in  pairs  are  taken,  and  all  the 
terms  added  after  each  is  separately  squared. 

Again,    2«-/3  =  a-/3  +  ay  +  (By  +  /?-«  +  y-«  +  y'(3, 

where  all  possible  permutations  of  the  roots,  two  by  two,  are 
taken,  and  the  first  root  in  each  term  then  squared. 
In  the  case  of  a  biquadratic,  we  have 

2«^/3-  =  «'/32  +  ay  +  a^W  +  fiY  +  [B-h-  +  y-8l 

We  give  a  few  examples,  which  may  serve  to  give  the  stu- 
dent some  insight  into  the  formation  of  this  class  of  functions. 


EXAMPLES. 

1.   Find  the  value  of  2«"/3  of  the  roots  of  the  cubic  equation 

or'  +  px^  -\-  qx  -\-  r  =  0. 
Multiplying  together  the  equations 

ySy  4-  y«  +  a(i  =  q, 
we  obtain  ^a-(3  +  3  «/3y  =  -  2)q ; 

hence  2«-/3  =  3  r  -  ^)g. 


Alt.  101  BOOTS  AND   COEFFICIENTS.  135 

2.  Find  for  the  same  cubic  the  value  of  a^  +  fi^  +  y-. 

Ans.  2«-  =  p^  —  2  q. 

3.  Find  for  the  same  cubic  the  value  of 

Multiplying  the  values  of  2«  and  2«-,  we  obtain 
«3  _^  ^3  +  ^3  _^  2(r/8  =  -i/  +  2iJ5; 
hence,  by  Ex.  1,       Sk"*  =  —  j>^  +  3  jxj  —  3  >•. 

4.  Find  for  the  same  cubic  the  value  of 

2«-)8-. 

5.  If  a,  (3,  y,  8  are  the  roots  of  the  biquadratic  equation 

X*  +ixr''  +  (J.,^  +  r.v  +  s  =  0, 
find  the  value  of  the  symmetric;  function 
2«2y8y  =  «-/8y  +  (ciS^  +  «'yS  +  l^'uy  +  /3'-'«8  +  /3-yS  +  y-u(3  +  y-(<8 
4-  y'/5S  -f  8'«/3  +  8-«y  +  S-^y. 
Multiplying  together 

a  +  (3  +  y  +  8  =  -p, 
apy  +  a(3B  +  «y8  +  /8y8  =  -  r, 
we  obtain  2«-^y  +  -1  «/3y8  =  i)r ; 

hence  2a^)8y  =  i^r  -  4  .<?. 

6.  Find   for  the  same  biquadratic  the  value  of  the  sym- 
metric function 

rr  +  /3'  +  y-4-8l 

7.  Find  the  value,  in  terms  of  the  coefficients,  of  the  sum 
of  the  squares  of  the  roots  of  the  equation 

a-»  +  pi^"'^  +  P-^"^"'^  4-  •••+;>„  =  f>- 

Ans.   la-"  =  ]>r  -  2 p.. 


CHAPTER  VIII. 

TRANSFORMATION   OF   EQUATIONS. 

In  many  cases  the  discussion  and  solution  of  an  equation 
is  facilitated  by  some  algebraic  transformation  that  will  change 
it  into  a  form  more  convenient  for  investigation.  We  shall 
now  consider  some  simple  and  useful  cases  of  transformation. 

102.  To  transform  an  Equation  into  Another,  the  Roots  of 
which  are  those  of  the  Proposed  Equation  with  Contrary  Sign. 

Let  «i,  u.,,  Us  •■•  "«  l^Je  the  roots  of  the  equation 

.r"  +  i>i;«'*-i  +  i^-C'-^H +})»  =  0. 

We  have  then  the  identity 

X"  +  pi.r"-^  +  2)-2X''--  +  .  •  •  +  p„ 

=  (.^  -  «i)  (X  -  Uo)  (x  -Us)---  (x  -  «„). 

Changing  x  into  —  y,  we  have,  whether  n  be  even  or  odd, 

2/"  -  2hU"~'  +  i\'2/""' T  Pn-iy  ±  P^ 

=  (y  +  «i)  (y  +  «2)  (y  +  «3)  •••(?/  +  «„)  =  o. 

The  roots  of  the  last  equation  are  —  «i,  —  Uo,  —  a^---  —  «„, 
and  thus  the  transformed  equation  may  be  obtained  from  the 
given  equation  by  changing  the  sign  of  the  coefficient  of  every 
other  term  beginning  icith  the  second. 

In  applying  this  rule  to  an  equation  that  is  not  complete, 
we  must  first  supply  the  missing  terms  by  writing  them  down, 
each  in  its  proper  place  with  zero  for  a  coefficient. 
136 


Art.  103  TRANSFORMATIOX    OF  EQUATIONS.  137 

EXAMPLES. 

1.  Find  the  equation  whose  roots  are  the  roots  of 

a;*'  —  4  X'  +  3x^  +  x^  +  7  jr  +  2x-i-5  =  0 
witli  their  signs  changed. 

2.  Change  the  signs  of  the  roots  of  the  equation 

cc»  +  2  a;«  +  4  x*  +  x'  +  o  x-  +  G  =  0. 

Ans.  x^  +  2x^  +  4:X*  —  x^  +  ox^  +  (j  =  0. 

103.  To  transform  an  equation  into  another,  the  roots  of  ichich 
are  equal  to  those  of  the  proposed  equation  multiplied  by  a  r/icen 
quantity. 

Let  «i,  a.2,  «3,  •••  «„  be  the  roots  of  an  equation  f(x)  =  0,  and 
let  it  be  required  to  transform  the  proposed  equation  into 
another,  the  roots  of  which  shall  be  ka^,  ka.^,  ku^,  •••  ka„. 

Assume  x  =  ■-,  and  substitute  in  the  identity  of  the  preced- 
Ic 
ing  article.     After  multiplying  by  k'\  we  have 

?/"  +  A-/)!^"-!  +  k^p--y"~-  +  •••  +  l^"~hh,-^I/  +  ^•"i>n 
=  (M-ku,)(y-ka.;)-'(>/-ku„). 

Hence,  to  multiply  the  roots  of  an  equation  by  a  given 
quantity  k,  we  have  only  to  multiply  the  successive  coefficients, 
beginning  ivith  the  second,  by  k,  k^,  k"^,  •••  k". 

Any  missing  power  of  x  must  be  written  with  zero  as  its 
coefficient  before  the  rule  is  applied. 

This  transformation  is  very  useful  for  removing  the  coeffi- 
cient of  the  first  term  when  it  is  not  unity,  and,  in  general, 
for  removing  any  fractional  coefficients.  When  thi-re  are  frac- 
tional coefficients,  we  get  rid  of  them  by  using  a  multiplier  k 
which  may  be  determined  by  inspection. 


138  THEORY  OF  EQUATIONS.  Art.  103 

EXAMPLES. 

1.  Change  the  equation 

2  x^  -  3  .v"  +  5  or^  -  4  X  +  6  =  0 

into  another  the  coefficient  of  whose  highest  term  will  be  unity. 
We  jnultiply  the  roots  by  2. 

Ans.  x'-3  x"  +  3  0  .ar  -  16  x  +  48  =  0. 

2.  IMake  a  similar  transformation  for  the  equation 

3  x^  +  a;^  -  5  ic^  +  2  ar  -  7  a;  +  5  =  0. 

3.  Kemove  the  fractional  coefficients  from  the  equation 

x'-\x''  +  ^x-l  =  Q. 

Here  we  multiply  the  roots  by  6,  thus 

a.-3-i(6)x^  +  |(r,)-\'«-(6>^=:0. 

Am.  x^  -  3 ;«-  +  24  a;  -  21 6  =  0. 

4.  Remove  the  fractional  coefficients  from  the  equation 

supply  missing  term,  and  use  10  as  a  multiplier. 

Ans.  X*  +  30  a;2  +  520  a;  +  770  =  0. 

Eemove    the    fractional     coefficients    from    the    following 
equations : 

5.  a;*  -  y2_ x-  +  ^x  +  l  =  0. 

6.  cc3-|a;2  + |a;-|  =  0. 

7.  a-3-ix^-J^.T  +  ^i3=0. 

-  8.   x'-\x^-^lx'+\x--,l,^^0. 


Art.  105  TRANSFOliMAriOX  OF  EQUATIOyS.  131> 

104.    To    Transform  an  Equation  into    Another  the  Roots  of 
which  are  the  Reciprocals  of  the  Roots  of  the  Proposed  Equation. 

Here   we   substitute  -  fur  x  in   the  identity  of   Article   1(12. 

!f 
Making  this  substitution  and  reducing,  we  have 

i  Pi  P>  Pn     1 

yn  yn-l  yU        .  ^  ^     " 

Hence,  if,  in  the  given  equation,  we  replace  x  by  -  and  naul- 

y 

tiply  by  y",  the  resulting   equation  will   have  for   roots  the 
reciprocals  of  «i,  Uo,  •••  «„. 


EXAMPLES. 

Find  the  equations  Avhose  roots  are  the  reciprocals  of  the 
roots  of 

Alls.  2y*-5f-7y^  +  3y-l  =  0. 

2.  x'-T  x"  +  4  .r-  -  7  a;  +  2  =  0. 

3.  x«  -  5  x'  -  or'  +  5  x2  +  7  X  +  10  =  0. 

4.  a^-'Sar-6  =  0. 

105.   Infinite  Roots.     If  j\  =  0,  one  root  of  /(x)  =  0  is  zero, 
and,  therefore,  by  Art.  104,  the  corresponding  root  of 


^Q-'^J 


140  THEORY  OF  EQUATIONS.  Art.  105 

That  is,  if  in  an  equation  the  coefficient  of  a;"  (the  highest  poiver 
of  x)  is  0,  one  root  is  infinity. 
Thus,  one  root  of  the  equation 

{m  -  n)  X'-  3  mx-  +  2  a;  -  10  =  0 

is  infinite,  if  m  =  n. 

In  like  manner,  if  the  coefficients  ofx"  and  o;""^  are  both  0,  two 
roots  are  infinity,  and  so  on. 

106.  Reciprocal  Equations.  Reciproccd  or  recurring  equations 
are  tliose  which  remain  unaltered,  when  x  is  changed  into  its 
reciprocal. 

The  conditions  that  must  hold  among  the  coefficients  of  an 
equation  in  order  that  it  should,  belong  to  this  class  are,  by 
Art.  104,  as  follows : 

l)n-l  Pn-2  Pi  1 

— =p„—=p,  ...-=„„_„ --=j,.. 

The  last  of  these  conditions  gives  p^  =  1,  or  p,^  =  ±  1. 
Eeciprocal  equations  are  divided  into  two  classes,  according  as 
Pn  is  equal  to  +  1,  or  to  —  1. 

(a)    In  the  first  case,  we  have 

Pn-l  =  Ih,   Pn-2  =P2,     "  "    Pi  =  Pn-l  ', 

and  these  relations  determine  the  first  class  of  reciprocal  equa- 
tions, in  which  the  coefficients  of  the  corresponding  terms 
taken  from  the  beginning  and  end  are  equal  in  magnitude, 
and  have  the  same  sign. 

(h)   In  the  second  case,  when  p„  =  —  1,  we  have 

Pn-l  =  ~  Pi.    Pn-2  =  -P2,     "•    Pi  =  - Pn-l', 

which  relations  give  the  second  class  of  reciprocal  equations,  in 
which  corresponding  terms  taken  from  the  beginning  and  end 


Art.  lOG  TRANSFOTiMATIOX   OF  EQUATIOXS.  Ml 

are  equal  in  magnitude,  but  different  in  sign.  In  thi.s  case, 
when  the  degree  of  the  equation  is  even,  say  n  =  2  m,  one  of 
the  conditions  becomes  j)„  =  —  p„.,  or  p,„  =  0,  so  that  in  recipro- 
cal equations  of  the  second  class,  whose  degree  is  even,  the 
middle  term  is  absent. 

It  is  evident  that  tlie  roots  of  reciprocal  etiuations  occur  in 

pairs,  «,  - ;   I3,~;  etc.     Wlien  the  degree  is  odd,  there  must  be 
a         (3 

a  root  which  is  its  own  reciprocal,  and  it  is  obvious  that  in 

this  case  —  1  or  +  1  is  a  root  according  as  the  equation  is  of 

the  first  or  second  class.     In  either  case  we  can  divide  by  the 

known  factor  (x  +  1  or  x  —  1),  and  what  is  left  is  a  reciprocal 

equation  of  even  degree  and  of  the  first  class. 

In  equations  of  the  second  class  of  even  degree  .r^  —  1  is  a 
factor,  and,  by  dividing  by  .i-^  —  1,  this  is  reducible  to  a  recijv 
rocal  equation  of  the  first  class  of  even  degree.  Hence  all 
reciprocal  equations  may  be  reduced  to  those  of  the  first  class 
of  even  degree,  Avhich,  therefore,  may  be  regardeil  as  the  stand- 
ard form  of  reciprocal  equation.^. 

We  append  a  few  examples,  with  some  hints  as  to  the  method 
of  solving  such  equations. 

EXAMPLES. 

1.    Solve  the  reciprocal  equation 

X*  +  x^-i  X-  +  x-\-l  =  0. 
Dividing  through  by  x-,  this  becomes 

x-  +  x-4:  +  -  +  \  =  0. 

X        XT 

Adding  and  subtracting  2,  this  may  be  put  in  the  form 


and  (.  +  iJ+(.+  l)  +  Uf: 


142  THEORY  OF  EQUATIONS.  Art.  106 

therefore  x-\ 1--  =  ±-. 

X     2         2 

...  ;,;  +  ?:^2or  -3. 
a; 

Solving  this  quadratic,  the  first  value  gives  x  =  l,  and  the 

1     •                —  3  ±  VS 
second  gives  x  — 


2.    Solve  the  equation 

x^-l  =  0. 

This  is  a  reciprocal  equation  of  the  second  class.  Dividing 
by  X  —  1  (since  x  =  1  is  evidently  a  root),  we  reduce  it  to  the 
reciprocal  equation  of  the  first  class  of  the  fourth  degree. 

x'^  +  x^  +  x'  +  if  +  1  =  0, 

or,  dividing  by  x^  and  arranging  terms, 

Therefore  (a^'  +  H  +(x  +  -]=l. 


Solving  this  as  in  the  preceding  example,  we  get  finally 
a-  =  ill  T  V5  ±  V^=T(10  ±  2  V5)^|, 
which  expression  gives  the  four  values  of  x. 

3.  Keduce  to  a  reciprocal  equation  of  even  degree  and  of 
first  class 

a;«  +  1^5  _  2_2  ^4  _,_  2^2  a,2  _   5  3.  _  1  =  0. 

4.  Solve  the  reciprocal  equation 

2a;6  +  a:^  _  i3_^4  ^_  13^,2  _  3.  _  2  =  0. 

Divide  the  left-hand  member  by  x"^  —  1. 


0 


Alt.  107  TliANSFOUMATlOX    OF  EQUATIOXS.  113 

107.  To  transform  an  Equation  into  Another,  the  Roots  of 
which  shall  be  Less  ( or  Greater;  than  those  of  the  Proposed  Equa- 
tion by  a  Constant  Difference. 

Let  /(.1-)  =  0  be  the  proposed  equation.  In  this  equation 
we  change  x  into  y  +  k.  The  resulting  equation  in  y  will  have 
roots  each  less  or  greater  by  k  than  the  given  equation  in  x, 
according  as  k  is  positive  or  negative.  The  resulting  equation 
is  (Art.  SO) 

/W +/(A-)y  +  4^^  /  +  -  +  3/"  =  0. 

The  following  mode  of  formation  of  this  equation  is,  for 
practical  purposes,  much  more  convenient  than  the  direct  cal- 
culation of  the  derived  functions  and  the  substitution  in  them 
of  A-. 

Let  the  proposed  equation  be 

f(x)  = .-«"  +  2)iX"-^  +  2)-2X"-^  +  •  •  •  -f  Pn-ix  -f  2\  =  0, 
and  suppose  the  transformed  polynomial  in  y  to  be 

PoU"  +  P^f-'  +  P-2U"-'  +    -    +  Pn-lV  +  Pni 

since  y  =:x  —  k,  this  is  equivalent  to 

p,(x  -  ky  +  p,(x  -  ky-'  -f  ...  -h  P„_,(.r  -  A-)  +  P„ 

which  must  be  identical  with  the  given  polynomial.  AVe  con- 
clude that  if  the  given  polynomial  be  divided  by  x  —  k,  the 
remainder  is  P,„  and  the  quotient 

p,{x  -  ky-'  +  p,(x  -  ky-'  +...-}-  P,._o(x  -  k)  +  p,._, ; 

if  this  again  be  divided  by  a;  —  A",  the  remainder  is  P„_i,  and 
the  quotient 

p,(x  -  ky-'  +  p,(x  -  ky-''  +  "'  +  /'„_2. 

Proceeding  in  this  way  we  can,  by  a  repetition  of  the  opera- 
tions explained  in  Art.  82,  calculate  in  succession  the  several 


144 


THEORY  OF  EQUATIONS. 


Art.  107 


coefficients  P,„  P„_i,  etc.,  of  the  transformed  equation ;  the 
last,  Po,  being  equal  to  unity,  as  we  know  from  other  con- 
siderations. 

We  shall  find,  when  we  give  in  Chapter  IX.  an  explanation 
of  Horner's  Method,  that  the  best  practical  method  of  solving 
numerical  equations  is  only  an  extension  of  the  process  here 
indicated.     A  few  examples  will  make  the  process  plain. 

EXAMPLES. 
1.   Find  the  equation  whose  roots  are  the  roots  of 
a;''  +  x3-29ic2-9ic  +  180, 
each  diminished  by  6. 

The  calculation  is  best  exhibited  as  follows : 


1    1 

-29 

-9 

180 

6 

42 

13 

78 

91 
114 

78 

69 
546 

414 

7 
6 

594 

13 
6 

615 

19 

205 

6 

25 


Here  the  first  division  of  the  given  polynomial  by  cc  —  6 
gives  the  remainder  594  (P4),  and  the  quotient 

a;^  +  7  X-  +  13  .T  +  69  (compare  Art.  80). 

Dividing  this  again  by  x  —  6,  we  get  the  remainder  615  (P.^) 
and  the  quotient  x^+13x  +  91.  Dividing  again,  we  get  the 
remainder  205  (P3)  and  the  quotient  x  +  19,  and  dividing  this 


Art.  107  TRANSFOUMATWX    OF  EQUATIOys.  145 

we  get  Pi  =  25,  and  1^  =  1;  hence  the  recpiii-ed  transformed 
equation  is 

i/  +  2oif  +  205  y-  +  G15  f/  +  594  =  0. 

2.  Find  the  equation  whose  roots  are  the  roots  of 

af +  4ar'-x2  +  ll  =  0, 
each  diminished  l>y  3. 

Ans.  f  +  loy'  +  di7f  +  305 y-  +  507 y  +  353  =  0. 

3.  Find  the  equation  whose  roots  are  the  roots  of 

4  x^  -  2  ar^  +  7  a;  -  3  =  0, 
each  increased  by  2. 

Here  we  divide  by  x  +  2,  as  follows : 


0-2 
8  16 


0 


7 
56 


-     3 
-126 


-  8 

14 

-  28 

63 

-129 

-  8 

32 

-  92 

240 

-16 

46 

-120 

303 

-  8 

48 

-188 

-24 

94 

-308 

-  8 

64 

-32 

158 

-  8 

-40 
The  transformed  equation  is  therefore 

4  ^  _  40  7/^  +  158  /  -  308  /  +  303  y  -  129  =  0. 

4.  Increase  by  5  the  roots  of  the  equation 

3x*  +  7ar''-15.r  +  J^-2  =  0. 

5.  Diminish  by  20  the  roots  of  tlie  equation 

r,y._  i:5.,^_  12x4- 7=0. 


146  THEORY   OF  EQUATIONS.  Art.  108 

108.  Removal  of  Terms.  The  solution  of  an  equation  is 
often  facilitated  by  the  removal  of  a  certain  specified  term, 
which  can  be  done  by  the  transformation  of  Art.  107,  as  we 
shall  now  show. 

If  f{x)  =  0  be  expressed  in  the  form 

ci()X"-  +  ciiX"'^  +  a^x"^^  -}-...  +  a„  =  0, 

and  the  transformed  equation  be  written  in  descending  powers 
of  y,  we  have 

chVn  +  {nuok  +  a,)  y""^  +  j  ^^^."9      "'^'^  +  ^'^  ~  -^)  "1^"  +  ^*2  j-  2/""^ 

+  ...  =  0. 

If  we  give  Jc  such  a  value  that  naok  +  aj  =  0,  the  transformed 
equation  will  be  wanting  in  the  second  term. 

If  k  be  either  of  the  values  which  satisfy  the  equation 

n(7i  —  1)     .„      ,        ^ .     ,  r. 

— ij — -^y^  aji-  +  (w  —  1)  a-ji  +  cu  =  0, 

the  transformed  equation  will  Avant  the  third  term. 

To  remove  the  fourth  term,  a  cubic  equation  will  have  to  be 
solved;  and  so  on.  The  following  examples  will  illustrate  the 
method : 

EXAMPLES. 

1.  Transform  the  equation 

x^-6  X-  +  12  a:  +  19  =  0 

into  one  wanting  the  second  term. 

?m,/c  +  ai  =  0  gives  k  —  2;  therefore  we  must  diminish  the 
roots  by  2.  A7}S.  f  +  21^  0. 

2.  Transform  the  equation 

a;^_4.r"^-18a;2-3a;  +  2  =  0 
into  one  wanting  the  third  term. 


Art.  109  TRAXSFOIiMATION    OF  EQUATIOys.  147 

The  quadratic  for  k  is 

6  k-  -12k-  18  =  0,  giving  ^-  =  3,  ^  =  -  1. 

Thus  there  are  two  ways  of  effecting  the  transformation. 
Diminishing  the  roots  by  3,  we  get 

y'  +  Sf-iny-19(J  =  0. 

Increasing  the  roots  by  1,  we  get 

y^-Sf  +  17y-S  =  0. 

3.  Transform  the  equation 

x'-\-Sx^  +  x-5=0 
into  one  wanting  the  second  term. 

4.  Transform  the  equation 

x^-6x'  +  9x-10  =  0 
into  one  wanting  the  third  term. 

109.   The  Algebraic  Solution  of  the  Cubic  Equation.     Let  the 

general  cubic  equation  he  written  in  the  form 

x'  +  Sp,x-  +  Si^^x  +  p^  =  0 (1) 

We  first  simplify  this  by  transforming  it  into  an  equation 
lacking  the  second  term.  To  do  this,  we  replace  x  by  y  +  k 
(Art.  107),  where  k  is  determined  by  the  equation  (Art.  108) 

3^•  +  37),  =  0, 
which  gives  k  =  —  j). 

Then  (1)  becomes 

iy-ihf  +  ^ih{y-ihf-\-^ih(.y-ih)+ih  =  ^    •    (2) 

which  reduces  to  the  form 

if  +  3IIy  +  G  =  0 (3) 

where  H  =  ih—  Pi   and  G  =  2  p^  -  3  ;>,  p.,  +  ;>,. 


148  THEORY  OF  EQUATIOXS.  Ai't.  109 

To  solve  (3),  assume 

y  ^j.i  +  si; 

...  t/3-3?-y?/-(r  +  .s)  =  0 (4) 

Comparing  coefficients  in  (3)  and  (4),  we  have 
o^h^  =  -H,  r+s  =  -G; 
from  which  equations  we  obtain 

r  =  |(-G=+VG^TT^) (5) 

s  =  ^(-G-VW+Tii^) (6) 

J  TT 

and,  substituting  for  s^  its  value  — —,  we  have 

y  =  r^  +  ^ (7) 

9-3 

the  value  of  r  being  given  in  (5). 

AVe  observe  that  if  r  be  replaced  by  s,  this  value  of  y  is 
unchanged,  as  the  terms  are  then  simply  interchanged;  also, 
since  r^  has  the  three  values  -Vr,  m-Vr,  w^vV,  obtained  by  mul- 
tiplying any  one  of  its  values  by  the  three  cube  roots  of  unity 
(Art.  100),  we  obtain  three,  and  only  three,  values  for  ?/;  namely, 

We  have  then  x+2h  =  '>'^  +  ^^ (8) 

as  the  comi^lete  algebraic  solution  of  the  cubic  equation 

ar'  +  S2hx-  +  3  j72X  +  2h  =  0, 

the  square  root  and  cube  root  involved  being  taken  in  their 
entire  generality.* 

*  This  solution  is  known  as  Cnrdan\  Solution,  because  it  was  first  published  by  him  in 
IW!).    See  Uistorical  Note,  page  76. 


Art.  no  TRANSFORMATION   OF  EQUATrOXS.  149 

110.  Application  to  Numerical  Equations.  The  solutiuu  of 
the  cubic-  ubtaiiird  in  the  last  artirk-  is  of  little  practical  value, 
wlien  the  equation  has  numerical  coefticieut.s.  For,  when  the 
roots  are  all  real  and  unequal,  G' -\- A  IP  <_  0  (this  may  i>e 
shown  by  Sturm's  Theorem,  see  Chapter  IX),  wlience  }•  is 
imaginary,  and  the  roots  involve  the  square  root  of  an  imagi- 
nary number,  Avhich  in  general  we  cannot  solve.  If  the  equa- 
tion has  equal  roots,  it  can  be  solved;  and  if  it  has  a  pair  of 
imaginary  roots,  it  likewise  can  be  solved,  for  in  this  case 
G'-  +  4  IP  is  positive.  In  the  first  case,  namely,  when  the 
roots  are  all  real,  the  roots  may  be  computed  by  the  use  of 
Trigonometry.* 

To  illustrate  this  method  by  an  example,  let  \is  solve  the 
equation 

a^-lSx-3o  =  0 (1) 

Put  x=r^  +  s^', 

...  ar  -  3  rM-K  -  (r -f  .s)  =  0 (2) 

.-.  7-y  =  6,  r  +  s  =  3o, 

r^  =  3,  s^  =  2 ; 


*  Throuphoiit  a  trontiso  „{  the  >.Ta<le  and  scope  of  this  work,  there  is  obWoiisly  much 
matter  that  must  be  left  iinnotieed.  It  woul.l  be  Intcrestlnir  to  jrlve  s.iine  tri«ronoinetric«l 
solutions  of  the  cubic  and  bi-.iuadmtic.  an.l  the  author  reluctantly  dUmissos  the  subject 
by  referring  the  student  to  more  extended  treall--'  •■"  ''i-  iii^orv  of  Ki)uutions. 


.-.  x  = 

-.r^  +  s^  = 

:3  +  2 

The  other 

two  roots  are 

w  V/' 

w  vr 

-t  + 

<J^Vr 

+  --'^  = 

^0,^'^, 

.■;  _ 

150  THEORY    OF    EQUATIONS.  Art.  110 

After  getting  the  real  root,  it  is  often  simpler  to  depress  the 
equation  and  tlien  get  the  two  imaginary  roots  by  solving  the 
resulting  quadratic.     Here  the  depressed  equation  is 

x'  +  Bx  +  l  =  0 (3) 

and  the  roots  of  this  quadratic  are 

5  ' 


which  agrees  with  what  we  have  just  obtained. 

EXAMPLES. 

Solve  the  following  equations : 

1.  x^  -  6a;-  +  10a;  =  8.  Ans.  4,  1  +  V^^,  1  -  V^^. 

2.  ar'  -  9  af  +  28  a;  =  30.  Ans.  3,  3  +  V^l,  3  -  V^=l. 

3.  ar^  + 72  a;  =  1720.  5.    a;^  _  g  a;^  +  13  a;  =  10. 

4.  .T^  +  63  a;  =  316.  6.    a.-^  -  6  a;^  +  3  a;  -  18. 

111.   Solution  of  the  Biquadratic  Equation.* 

Here  we  find  it  convenient  to  put  the  biquadratic  in  the  form 

a;*  +  2^9x3  +  fya;2  +  2  ra;  +  s  =  0 (1) 

Adding  (ax  +  by  to  both  members,  we  obtain 

x*-\-22JX^+(q+a'')x--\-2{r+ab)x+s+b^=(ax+by.     .     (2) 
Assume 

x*-{-22Jx^-\-(q-\-(r)x^+2{r+ab)x+s+b-=(x^-\-2-)x+ky.     .     (3) 
Equating  coefficients,  we  have 

p2  +  2k  =  q-\-a'^,  x>k  =  r  +  ab,  k- =  s  +  b-.     .     .     (4) 

*The  solution  here  given  is  due  to  Ferrari.  (See  Ilistoriciil  Note,  pajre  Tfi.)  This 
and  the  solutions  of  Descartes,  Euler,  Laplace,  Lajrraiitjc,  and  others,  all  involve  the  solution 
of  a  cubic  by  Cardan's  method,  and  will  of  course  fail  when  that  fails.  We  would  then 
employ  a  trigonometrical  solution. 


Alt.  Ill         TliANSFOIiMATION    OF    EQUATIONS.  lol 

Eliminating  a  and  b  from  (4),  we  have 

or  2A-•'-7^•^'4-20>r-s)^•^;)2^•f  7.s-/-2  =  0.     .     .     (5) 

From  this  cubic  we  find,  if  possible,  a  real  value  of  /,•  b}'  the 
method  of  Art.  109.  The  values  of  a  and  b  are  then  known 
from  (4). 

Subtracting  (2)  from  (3),  we  have 

(a^'+^,.v  +  7,)2_  (ax +  6)2=0, 

which  is  equivalent  to  the  two  quadratic  equations 

x"  +  (i)  -  a)  X  +  (k  -  6)  =  0, 

x'+(2^  +  a)x  +  (k  +  b)  =  0, 

the  roots  of  which  are  readily  obtained. 

As  an  example  of  this  method,  let  us  solve  the  equation 

;^.4  + 2.1'-' -7  a- -8.^  +  12  =  0 (1) 

Adding  (ax  +  b'f  to  both  members,  Ave  obtain 

x*+2  x'+(a'-7)x''+2(ab-4:)x+b' +  12  =  (cuv+b)-.     .     (2) 

Since  p  =  -\-l,  assume 

a'+2a^+(a'-7)x'+2(ab-4)x+b'+l2  =  (x'+x+ky     .     (3) 

Equating  coefficients,  Ave  have 

a'-T  =  2k  +  l,  ab  -  4  =  A-,  b'  +  12  =  A.-     .     .     (4) 

.-.  (2k  +  S)ik'-12)  =  (k  +  4y, 

...  2  Jc'  +  7k--  :V2  A-  -  112  =  0. 

Whence  k  =  4;  hence  a-  =  IH,  ab  =  8,  b'-  =  4,  and  .-.  a  =  4, 
6  =  2. 


152  THEORY  OF  EQUATIONS.  Art.  Ill 

Therefore,  from  (2),  (3),  and  (4),  we  olDtain 
(or' +  .T  + 4)2 -(4a; +  2)2  =  0, 
which  is  equivalent  to  the  two  equations 

ar  -  3  X  +  2  =  0,  a;^  +  5  .r  +  6  =  0 ; 
and,  therefore,  the  four  roots  are  1,  2,  —  2,  —  3. 

EXAMPLES. 

1.  Solve  X*  -  6  x"  -f  12  x^  -  14  x-  +  3  =  0. 

2.  Solve  X*  +  4  af  +  3  x'  -  44  .r  -  84  =  0. 

3.  Solve  a;*-6x-2-8x-3  =  0.        Ans.   -1,   -1,   -1,  3. 

4.  Solve  X*  -  3  x2  -  42  a;  -  40  =  0. 


r 


CHAPTER   IX. 

LIMITS    OF   THE   ROOTS    OF   AN   EQUATION. 

112.  Definition  of  Limits.  In  attempting  to  find  the  real 
roots  of  numerical  ecpiations,  it  is  very  advantageous  to  nar- 
row the  limits  within  which  such  roots  must  be  sought.  iJes- 
cartes'  Kule  of  Signs  gives  us  the  limit  of  the  number  of  real 
roots,  but  tells  us  nothing  as  to  the  limit  of  the  value  of  such 
roots.  The  closing  remarks  of  Art.  78  suggest  that  there  are 
means  of  getting  the  limits  between  which  the  roots  of  a  given 
equation  must  lie,  and  we  shall  now  proceed  to  give  some 
of  the  methods  for  doing  this. 

A  superior  limit  of  the  j^ositice  roots  is  any  positive  number 
greater  than  the  greatest  of  the  roots,  that  is,-  nearer  +  x ; 
an  inferior  limit  of  the  positive  roots  is  any  positive  number 
smaller  than  the  smallest  of  them. 

A  superior  limit  of  the  negative  roots  is  any  negative  number 
greater  in  absolute  value  than  the  greatest  of  them,  that  is, 
nearer  to  —  co  than  the  greatest ;  an  inferior  limit  of  the  nega- 
tive roots  is  any  negative  number  smaller  in  absolute  value 
than  the  smallest  of  them.  In  the  next  three  articles  we  liave 
three  rules  for  the  determination  of  the  superior  limits  of  the 
positive  roots. 

113.  PiioposiTiox  I.     ///  an  equation 

f(x)  =  .V"  -f  />,.»•"-'  +  y^a-"--  +  •••  +  7',,-i-r  +  Pn  =  0, 

if  the  first  negative  term  he  —  p^c"'',  and  if  lite  greatest  negative 
coefficient  be  —  Pt,  then  ■\/pi,-\-\  is  a  superior  liuiit  of  the  pnsilive 
roots. 

163 


154  THEORY  OF  EQUATIONS.  Ait.  11 -J 

Now  fix)  is  certainly  positive  for  any  value  of  x,  which 
makes  a^ 

^.n-r+l  _  J         .',     f  y'     ^ 

X'"  >  Ihi^'^"'  +  iv""""^'  H \-x-\-l)>2h ' z — 

X  —  1 

But  this  inequality  is  true,  taking  a;  >  1,  if 

X  —  1 

or  a;"+^  —  cc"  >  Pi^""''"'"^ 

or  a;  —  1  >  jJ^.*"''"^^, 

or  aj'  —  a?*""^  >  jikj 

that  is  a;'-'(a;  —  1)  >  p^. 

Bat,  since  a;''"^  >  (x  —  1)''~\ 

a;'-'(a;  —  1)  is  >  j)^,  if  (x  —  iy-'^{x  —  1)  >p^, 
or  (a;-l)'->j>,. 

Hence  /(a;)  will  always  be  positive,  if  a;  =  or  >  1  +  Vp.t. 

Hence  A/j^t  +  1  is  a  superior  limit  of  the  positive  roots. 

114.  Proposition  II.  If  in  any  equation  each  negative 
coefficient  he  taken  positively,  and  divided  by  the  sum  of  all  the 
positive  coefficients  ivhich  precede  it,  the  greatest  quotient  thus 
formed  increased  by  unity  is  a  superior  limit  of  the  positive 
roots. 

Let  the  equation  be 

OoX"+aia;''-^+  a^a;"-'-^-  a.^x"'^-\ a,a;"-'■^ [-  «„=  0     (1) 

in  which  we  regard  the  fourth  coefficient  as  negative,  and  we 
consider  also  a  general  negative  coefficient ;  namely,  —  «,.. 
Now,  since 

a;™—  1 

=  x'"-' +  a;"'-^ -^ \-x-\-l, 

X  —  1 

we  have     a;""  =  (x  —  1)  (x"-'  +  x""-^  + 1-  .x  +  1)  -f  1. 


Art.  114  ROOTS   OF  AN   EQUATIoy.  l.");> 

Let  us  now  develop  each  positive  term  of  equation  (1)  by 
the  formula 

o„.x'"  =  «„.(-^'  -  1)  (.^•'""^  +  x""'-  H \-x  +  l)+  a„, 

the  negative  term  remaining  uut-hanged. 

The  polynomial /(j;)  becomes  then:  Oq (^  —  1 ) .r""* 

+  ao(x  -  l)a;"-2+  a^ix  -  l)x^-'^-\ h  ao{x  -  l)x'-'+  •■•+a„, 

+  Oi(.c  -  l)a;"-2+  ai(x  —  \)x"-^-\ f-  a^{x  —  l)x"-^-\ h  a„ 

+  a.2(x  —l)x"  ^+  •••  +  a./x  —  1) j;"~''4-  •••  +  a,, 


a^x     , 

+ 


In  the  new  polynomial  thus  formed,  representing  the  left- 
hand  member  of  the  transformed  equation,  the  successive  co- 
efficients of  x"~\  a;"  ^,  etc.,  are 

ao(x  —  1),  («o  +  tti)  (x  —  i),  («o  +  «i  +  Ol>)  (^*  —  1)  —  «3'  etc. 

Any  value  of  x  greater  than  unity  is  sufficient  to  make  posi- 
tive every  term  in  which  no  negative  coefficient  O;,,  o^  etc., 
occurs.     To  make  the  latter  terms  positive,  we  must  have 

(ao  +  ai  +  «,)  (x  -  1)  >  (h, 


(tto  -f  Cfi  +  «2  +   •••  «r-l)  (•»  —  1)  >  «r>    etc. 


Hence  a;  >  — ; — ^ h  1 

tto  +  «!  4-  "2 


a;> 1-1,  etc. 

a„  +  rti +  «.+  •••  4- a,_i 


156  THEORY  OF  EQUATIONS.  Art.  114 

If  now  we  take  for  x  the  greatest  of  all  these  quantities,  the 
first  member  will  be  positive  (for  this  value  and  for  all  greater 
values  of  x) ;  and  this  will  be  a  superior  limit  of  the  roots. 

115.  Limit  obtained  by  grouping  Certain  Terms.  It  is  usually 
possible  to  determine,  by  inspection,  a  limit  closer  than  that 
given  by  either  of  the  preceding  propositions.  In  this  method 
we  arrange  the  terms  of  an  equation  in  grcmps  having  a  posi- 
tive term  first,  and  then  observe  what  is  the  lowest  integral 
value  of  X,  which  will  have  the  effect  of  rendering  each  group 
positive.     Such  a  value  of  x  will  be  a  superior  limit  of  the  roots. 

The  form  of  the  equation  will  suggest  the  arrangement  into 
groups  in  each  case. 

Of  the  propositions  in  the  two  preceding  articles,  sometimes 
one  will  give  the  closer  limit,  sometimes  the  other.  In  most 
cases  Prop.  II  will  give  the  closer  limit.  Of  course  the 
smaller  the  number  found,  the  better.  We  consider  the  inte- 
ger next  above  the  numerical  value  found  by  either  rule  as  the 
limit. 

EXAMPLES. 

1.  Find  a  superior  limit  of  the  positive  roots  of  the  equation 

x*-5a^  +  A0x^-Sx  +  23  =  0. 

Art.  113  gives  8  +  1,  or  9,  as  a  limit,  Art.  114  gives  |^  +  1,  or 
6,  as  a  limit. 

Hence  6  is  a  superior  limit. 

2.  Find  a  superior  limit  of  the  positive  roots  of 

x^  +  4  x''  -  3  x'  +  5 .«"  -  9  x-^  -llx'  +  6x-8  =  0. 

Art.  113  gives  5  as  limit. 
Of  the  fractions 

3  9  11  8 

1+4'    1+4  +  5'    1+4  +  A'    1+4  +  5  +  6' 

tlie  third  is  the  greatest,  and   Art.  114  gives  the  limit  3.     In 
this  case  Art.  114  gives  the  closer  limit. 


Art.  115  nOOTS   OF  AX  FAjU AVION.  157 

3.  Find  the  superior  limit  of  the  positive  roots  of 

x-'  +  8  x*  -  14  .it'  -  53  .ir  +  56  x  -18  =  0. 

Here,  Art.  113  gives  9  as  a  limit,  ami  Art.  114  gives  7  as  a 
limit. 

4.  Find  the  superior  limit  of  the  positive  roots  of 

x^  +  20  X'  +  4  .f«  -  11  x-^  -  120  X*  + 13  X  -  25  =  0. 

The  methods  of  Arts.  113,  114  both  give  the  limit  G. 
In  this  case  we  can  find  a  much  closer  limit  by  ajtplying  the 
method  of  Art.  115. 

The  equation  may  be  arranged  as  follows : 

X'  (ar'  - 11)  +  20  a-*  {t?  -  6)  +  4  .i-«  +  13  .r  -  25  =  0. 

Here  a;  =  3,  or  any  greater  number,  renders  each  group  posi- 
tive ;  hence  3  is  a  limit. 

5.  Find  a  superior  limit  of  the  roots  of  the  equation 

a;*  -x^-2  y?  -  4  .K  -  24  =  0. 

"When  there  are  several  negative  terms,  and  the  coefficient 
of  the  highest  term  is  unity,  it  is  convenient  to  multiply  the 
whole  equation  by  such  a  number  as  will  enable  us  to  distribute 
the  highest  term  among  the  negative  terms.  Here,  multiply- 
ing by  4,  we  can  write  the  equation  as  follows : 

r^{x  -  4)  +  .^-'  i?r  -  8)  4-  .^•  ix"  -  10)  +  x'  -  90  =  0, 

and  4  is  a  sujjerior  limit. 

Find  a  superior  limit  of  the  positive  roots  of  the  following 
equations : 

6.  4  ar*  -  8  .t:^  +  22  x^  +  98 .» '  -  73  .r  -f  5  =  0. 

7.  5  or^  -  7  x'  -  10  x^  -  23  x^  -  90  x  -317=  0. 

8.  ar>-a-*-2,,'''-|-2.r-f-a'-l=0. 

9.  x'*-8jr'  +  12.t;-  +  lGx-39  =  0. 


I 


158  THEORY  OF  EQUATIONS.  Art.  116 

116.   Inferior  Limits,  and  Limits  of  the  Negative  Roots. 

To  find  an  inferior  limit  of  the  positive  roots,  we  must 
transform  tlie  equation  into  another  whose  roots  are  the  recip- 
rocals of  those  of  the  first  by  the  substitution  x  =  -  (Art.  104). 

Find  then  the  superior  limit  I  of  the  positive  roots  of  the 
equation  in  y.     The  reciprocal  of  this,  -,  will  be  the  required 

inferior  limit ;  for  since  y  <I,  ->-,  i.e.,  a;  >  -. 
y      I  I 

Yov  example,  take  the  equation  of  example  (3)  under  the 

last  article 

a^  +  Sx*-Ux^-53x-  +  56x-lS  =  0.        .     .     (1) 

Putting  x=  -,  (1)  becomes 

y 

f-Hy'  +  nf  +  \V/--j%y-T\--(^,    •   .    (2) 

and  a  superior  limit  of  (2),  by  Art.  114,  is  ■&!  + 1  =||,  and, 
therefore,  if  is  an  inferior  limit  of  the  positive  roots. 

To  find  limits  of  the  negative  roots,  we  have  only  to  trans- 
form the  equation  by  the  substitution  x  =  —  y. 

This  transformation  (Art.  102)  changes  the  negative  into 
positive  I'oots.  If  I  and  V  be  the  superior  and  inferior  limits  of 
the  positive  roots  of  the  equation  in  y,  then  —  I  and  —  /'  are 
the  limits  of  the  negative  roots  of  the  proposed  equation. 

For  example,  take  the  equation 

x*-2x'-13x''-Ux-\-24:  =  0.       .     .     .     (1) 

Putting  X  =  —  y,  this  becomes 

y'-\-2f-13y'-  +  Uy  +  24.  =  (i.       ...     (2) 

By  the  method  of  Art.  115,  we  readily  find  a  superior  limit 
of  the  positive  roots  of  (2)  to  be  5 ;  therefore  —  5  is  a  superior 
limit  of  the  negative  roots  of  equation  (1). 


Alt.  117  ROOTS   OF  AX  EQUATION.  loO 

EXAMPLES. 
>^    1.    Find  limits  to  the  positive  and  negative  roots  of 

a;«  _  5  x^  +  x'  +  12  .»;■'  -  12  x-  +  1  =  0. 

Show  that  the  real  roots  of  the  following  eqnatious  lie  be- 
tween the  limits  respectively  given : 

V     2.    x^-a-3_(_4x-'-3.f  +  l  =  0;  ^  andl. 

3.  X*  +  o:'^  -  10  .IT  -  a-  +  15  =  0 ;   -  4  and  3. 

4.  •  ar>  +  5  x*  +  .^•'  -  10  ar  _  20  .r  -  IG  =  0 ;   -  5  and  3. 

5.  (x-  --ix-  2)-  -  43  =  0 ;   -  2  and  6. 

6.  x^  +  2  x^  +  3 .1-3  +  4  .r  +  5  x-  -  54321 ;   -  3,  9. 

Separation  of  the  Eoots  of  Equations. 

117.  Having  found  the  limits  within  which  the  real  roots 
of  an  equation  lie,  the  next  step  in  the  solution  of  an  equation 
is  to  discover  the  intervals  in'  which  the  separate  roots  lie. 
The  two  most  useful  theorems  for  determining  the  number 
of  real  roots  between  any  two  arbitrarily  assumed  values  of 
the  variable  are  the  Theorem  of  Fourier  and  Buchiu,  and  tlie 
Theorem  of  Sturm. 

For  a  proof  of  the  first,  we  refer  the  reader  to  Buru.side  and 
Paiito)i's  Tlieonj  of  Equations.  The  theorem  of  Sturm,*  wliich 
Ave  shall  consider  in  the  next  article,  has  the  advantage  of 
being  unfailing  in  its  application,  giving  always  tlie  exact 
number  of  real  roots  between  any  two  proposed  quantities; 

*  .1.  C.  F.  Sturm  (lS<«-IsV.). 


160  THEORY   OF   EQCfATIONS.  Art.  117 

whereas  the  theorem  of  Fourier  and  Ijudan  gives  only  a  cer- 
tain limit  which  the  number  of  real  roots  in  the  proposed 
equation  cannot  exceed. 

118.   Sturm's  Theorem.     Let 

f(x)  =  a-"  +  2hx''-'  +  •  •  •  +  2\-iX  +Pn  =  0  .     .     .     (1) 

be  an  equation  from  which  the  multiple  roots  have  been  re- 
moved (Art.  98).* 

To  find  the  equal  roots  we  have  employed  the  common 
operation  of  finding  the  H.  0.  F.  of  a  polynomial  f(x),  and 
its  first  derived  function,  f'(x).  Sturm  has  employed  the 
same  operation  for  forming  the  auxiliary  functions  which 
are  iised  in  this  method  for  separating  the  root's  of  an 
ecpiation. 

Let  the  process  of  finding  the  H.  C.  F.  of  f{x)  and  f'{x)  be 
performed. 

The  successive  remainders  will  go  on  diminishing  in  degree, 
and,  as  f(x)  has,  by  hypothesis,  no  multiple  roots,  f(x)  and 
f'(x)  have  no  common  divisor  except  unity,  and  we  finally 
()l)tain  a  remainder, /„(a;),  independent  of  x;  that  is,  which  is 
numerical. 

Dividing  f{x)  by  f'(x),  we  shall  obtain  a  quotient  f/j,  Avith  a 
remainder  of  a  degree  lower  than  that  of  f(x).  Denote  this 
remainder,  with  its  sign  changed,  by  /aO^),  and  divide  /'(j-)  by 
f-Ax),  and  so  on ;  the  operation  being  precisely  the  same  as 
lluit  of  finding  the  H.  C.  F.  of  f{x)  and  f{x),  except  that  the 
signs  of  each  remainder  must  be  changed,  Avhile  no  other 
changes  of  sign  are  permissible.  In  the  process  of  finding 
./i(-^)j  fi{^),  etc.,  any  j^ositive  numerical  factor  may  be  omitted 
or  introduced,  in  order  to  avoid  fractions,  for  the  sign  of  the 
result  is  not  affected  thereby. 

*  Tliis  limitation  is  not  neepssnry.  but  for  simplicity  we  consider  the  equation  cleared  of 
equal  roots,  as  this  can  always  be  done  by  the  method  of  Art.  DS. 


Art.  118  ROOTS   OF  AX   EQUATIOX.  IT.l 

The  expressions  f(x),  /'(x),  /^(x),  /,(x;  ■- f„(x)  are  called 
Sturm's  Finictious. 

Keeping  in  mind  the  above  explanations  and  definitions, 
we  may  now  state  Stunn's  Theorem : 

TiiKoKK.M.  //■  (Uiy  tico  real  numbers  a  mid  h  he  substituted 
for  X  ill  IStunns  Functions 

A^),  f{^),  M^)  -/.-.(^-j.  /nW, 

and  the  signs  noted,  the  difference  heticeen  the  number  of  chmirfes 
of  sign  in  the  series  tvhen  a  is  substituted  for  x,  and  the  vuniber 
when  b  is  substituted  for  x,  expresses  exactly  the  number  of  real 
roots  of  the  equation  f(x)  =  0  between  a  a)id  b. 

From  the  •way  in  which  Sturm's  Functions  are  formed,  we 
derive  the  following  series  of  equations,  in  which 

represent  the  successive  quotients  in  the  operation : 

f{x)^qj\x)-f{x) 
fiix)  =  q,f{x)-f(x) 

fr-i(^)  =  (JrfX-r)-fU^) 

/„-2(^-)  =  9,.-../;-i(-^)-./:.WJ 

Having  regard  to  these  relations,  we  observe : 

(1)  The  last  of  the  functions  f„(x)  is  not  zero;  for  by  sup- 
position it  is  independent  of  x,  and  if  it  were  zero,  the  equa- 
tion f(x)  =  0  would  have  equal  roots  by  Art.  1>S,  whii-h  is 
contrary  to  the  hypothesis. 

(2)  No  two  consecutive  functions  in  the  series  can  have 
a  connnon  factor;  for,  if  they  could,  all  the  succeeding  func- 
tions  would  vfttrrsh,  iHuluding   f„{^),  and    riiis    is   impossible 


162  THEORY  OF  EQUATIONS.  Art.  118 

(3)  When  any  auxiliary  function  vanishes,  the  two  adjacent 
functions  have  contrary  signs.  Suppose,  for  example,  that 
/^(■x)  =  0,  then  from  the  second  of  the  above  system  of  rela- 
tions we  have  /i(.t)  =  —fz{x). 

In  examining,  therefore,  what  changes  of  sign  can  take  place 
in  the  series  during  the  passage  of  x  from  a  to  h,  we  may 
exclude  the  case  i^f  two  consecutive  functions  vanishing  for 
the  same  value  (fK  the  variable ;  therefore  the  different  cases 
in  which  any  change  of  sign  can  take  place  are  the  following : 

(a)  When  x  passes  through  a  root  of  the  equation  f{x)  =  0. 

(6)  When  x  passes  through  a  value  which  causes  one  of  the 
functions  /',  f^,  /g  •••/„_!  to  vanish. 

(c)  When  x  passes  through  a  value  which  causes  two  or 
more  of  the  functions  /',  /a,  fs---fn-i  to  vanish  together;  no 
two  of  the  vanishing  functions,  however,  being  consecutive. 

(a)  When  x  passes  through  a  root  of  f(x)  =  0,  it  follows 
from  Art.  99  that  one  change  of  sign  is  lost,  since  immediately 
before  the  passage  f(x)  and  f'(x)  have  unlike  signs,  and  imme- 
diately after  the  passage  they  have  like  signs. 

(6)  Suppose  X  to  take  a  value  a  which  is  a  root  of  the  equa- 
tion /r(.i-)  =  0.     From  the  equation 

we  have  /r-i(«)  =  -/r+i(«)' 

which  proves,  as  we  have  seen,  that  this  value  of  x  gives  to 
/.  ,(.i')  and  fr+i(x)  the  same  numerical  value  with  different 
signs.  In  passing  from  a  value  a  little  less  than  a  to  one  a 
little  greater,  we  can  suppose  the  interval  so  sm^ll  that  it  con- 
tains no  root  of  X_i(a;)  or  fr+i{x) ;  hence,  throughout  the  inter- 
val under  consideration,  these  two  functions  retain  their  signs. 
^Ve  conclude  that  just  before  x,  varying  continuously,  reaches 
tlie  value  «,  the  signs  of  fr^i(x),  f^x),  fr+\{^)  must  be  -f  ±  — 
or  —  ±  +,  and  just  afterwards  they  must  be  +  T  —  or 
—  T  + ;  that  is,  f^{x)  changes  sign  as  x  passes  through  the 


Art.  119  ROOTS   OF  AN  EQrATloy.  KIS 

value  u,  and  the  other  two  do  not.  Biit^iough  the  sign  of 
X{-?0^ changeSj_no  Yarijitipn_jjf^^gn_^  is  eithei;jrqst  "or  gai ne'd^ 
thereby  in  the  group  of  three;  because,  on  ac-count  of  the 
difference  of  signs  of  the  two  extremes  ./j-il'*")  ^^^^^  /mi-'')' 
there  will  exist  both  before  and  after  the  passage  one  variii- 
tion  and  one  permanency  of  sign,  whatever  be  the  sign  of  the 
middle  function.  For  in  the  change  from  +  ±  —  to  -f-  T  — , 
or  from  —  ±  +  to  —  T  +,  a  permanency  and  a  variation  are 
changed  into  a  variation  and  a  permanency,  or  a  variation  and 
a  permanency  into  a  permanency  and  a  variation ;  but  no 
variation  of  sign  is  lost  or  gained  on  the  whole. 

(c)  It  follows  at  once  that  if  two  or  more  of  the  auxiliary 
functions  vanish  for  the  same  value  of  x,  since  no  two  adjacent 
ones  can  vanish,  the  same  reasoning  that  was  employed  in  {b) 
holds  good  here,  and,  therefore,  if  f(x)  is  one  of  the  vanishing 
functions,  one  change  of  sign  is  lost,  and,  if  not,  no  change  is 
either  lost  or  gained.  We  have  proved,  therefore,  that  when  x 
l>asses  through  a  root  of  f{x)  =  0,  one  change  of  sign  is  lost, 
and  under  no  other  circumstances  is  a  change  either  lost  or 
gained.  Hence  the  theorem :  the  number  of  changes  of  sign 
lost  while  X  varies  horn  a  to  i  is  equal  to  the  number  of  real 
roots  of  the  equation  between  a  and  b. 

119.  Separation  of  the  Real  Roots.  The  substitution  of  +  oo 
and  —  v:  for  .';  in  Stururs  Fuurtiuns  determines  the  number  of 
real  roots  of  f(x)  =  0. 

The  number  of  imaginary  roots  would,  of  course,  be  the 
difference  between  the  degree  of  the  equation  and  the  number 
of  real  roots  thus  determined.  The  substitution  of  -f-  x  and  0 
for  X  determines  the  number  of  positive  real  roots,  and  the 
substitution  of  —  oo  and  0  determines  the  numl)er  of  negative 
real  roots. 

In  applying  Sturm's  theorem,  it  is  convenient  in  practici'  tt> 
substitute  first  —  cc,  0,  +00  in  Sturm's  Functions,  so  as  to 
obtain  the  whole  number  of  negative  and  of   positive  roots. 


164  THEORY   OF  EQUATIONS.  Art.  119 

To  separate  the  negative  roots,  the  integers  —1,  —2,  —3,  etc., 
are  to  be  substituted  in  succession  till  we  reach  the  same  series 
of  signs  as  results  from  the  substitution  of  —  co ;  and  to  sep- 
arate the  positive  roots  we  svibstitute  1,  2,  3,  etc.,  till  the  signs 
furnished  by  +  co  are  reached.  A  few  examples  will  illustrate 
the  application  of  the  theorem. 

EXAMPLES. 

1.  Find  the  number  and  situation  of  the  real  roots  of  the 
equation 

/(a-)  =  x'-2x-5  =  0. 

We  find    /'  (x)  =  3  x'  -  2,  /.  (x)  =  4  .^  +  15,  f^  (x)  =  -  643. 
Corresponding  to  the  values  —  co,  0  +  co  of  x,  we  have 

(-^)   -    +    -    - 
(0)   -    -    +    - 

(+^)    +    +    +    - 
Hence  there  is  only  one  real  root,  and  it  is  positive. 
Again,  corresponding  to  values,  1,  2,  3  of  x,  we  have 

(1)  -    +    +   - 

(2)  -   +   +   - 

(3)  +   +   +   - 

The  real  root,  therefore,  lies  between  (2)  and  (3). 

2.  Find  the  number  and  situation  of  the  real  roots  of  the 
equation 

/(.r)  =  x'  -  6  x"  +  5  x'  +  14  X- 4:  =  0. 

Here  /'  (x)  =  2  or^  -  9  .-c-  +  5  x  +  7,  omitting  a  factor  2. 

f2{x)=:17x'-57x-5, 
f^{x)  =  152  X- 4:57, 

Mx)  =  +. 

In  this  exami)le  it  will  be  found   that   the   calculation   of 
fi{x)  is  somewhat  complicated;  it  is  sufficient  for  our  purpose^ 


Alt.  119  ROOTS   OF  AX  EQUATION.  165 

however,  to  know  the  aign,  and  thus  when  we  ascertain  that  it 
is  positive  we  need  not  calculate  it  exactly,  but  merely  put 
down  J\(x)  =  +.     Here  we  have  the  following  series  of  signs: 

(-co)    +    -    +    -    + 
(0)   -    +    -    -    + 

(+x)    +    +    +    +    + 

Hence  all  the  roots  are  real :  one  negative  and  three  positive. 
We  have  further  the  series  of  signs : 

(—2)  +  —  +  —  +,  4  variations. 

(— 1)—  —  +  —  +,  3  variations. 

(0)  —  +  —  —  +,3  variations. 

(1)  +  +  —  —  +,2  variations. 

(2)  +  —  —  —  +,  2  variations. 

(3)  +  —  —  —  +,2  variations. 

(4)  +  +  +  +  +,0  variations. 

There  is  one  change  of  sign  lost  between  —  2  and  —  1,  one 
between  0  and  1,  and  two  between  3  and  4. 

If  we  put  3|-  for  x,  the  succession  of  signs  is  —  0  +  +  +, 
and  thus  there  is  only  one  change  of  sign,  so  that  one  root  of 
the  equation  lies  between  3  and  3^;  therefore  another  root  lies 
between  3.V  and  4. 

Find  the  number  and  situation  of  the  real  roots  of  the 
equations : 

3.    a;^-3.r--4.T-f  13  =  0.  4.    x-''- 7x  + 7  =  0. 

5.  a;*-4ar'-3a;  +  23  =  0. 

Ans.  Two  real  positive  roots,  between  2  and  3,  and  3  and  4, 
respectively. 

6.  a;*-4ar''  +  ic2  +  6a;  +  2  =  0.  7.   a-* -|- .r-^  +  .r -  1  =  0. 
8.   ar'- 6x2 +  8 a; +  40  =  0. 


CHAPTER   X. 

ELIMINATION. 

120.  Under  the  head  of  Applications  of  Determinants,  in 
Chapter  III,  we  have  considered,  as  the  student  will  recall, 
several  cases  of  elimination  whereby  a  system  of  equations 
may  be  solved. 

In  Art.  41  there  was  given  the  method  of  solving  a  system 
of  simultaneous  equations  where  the  number  of  unknown 
quantities  is  the  same  as  the  number  of  equations. 

In  Arts.  42  and  43,  the  case  where  the  number  of  equations 
is  greater  than  the  number  of  unknowns  was  considered,  and 
the  condition  of  consistency  of  such  a  system  was  obtained.  In 
such  a  case  the  eliminant,  or  resultant,  which  is  the  determinant 
obtained  by  eliminating  the  unknowns  from  the  given  equation, 
is  the  determinant  of  the  coefficients  and  absolute  terms. 

We  next  considered  homogeneous  linear  equations  (Art.  44), 
and  found  that  for  a  system  of  w  homogeneous  linear  equations 
involving  n  unknowns  the  eliminant  is  the  determinant  of  the 
coefficients,  and  that  if  this  determinant  vanishes,  the  ratios  of 
the  unknowns  may  be  determined,  but  not  their  absolute  values. 

There  are  various  ways  of  determining  the  resultant,  R,  of  a 
system  of  equations.  We  shall  give  some  of  the  best  methods 
of  eliminating  a  single  unknown  from  two  consistent  equations 
of  any  degree. 

121.  The  method  that  naturally  presents  itself  is  as  follows : 
The  resultant  of  two  linear  equations 

ax  +  h  =  0,  a'a;  +  6'  =  0 

is  evidently  ab'  —  ba'  =  0. 

166 


Art.  122  ELIMINATION.  107 

If  now  we  have  two  quadratic  equations 
ax^  +  bx-\-c  =  Q     .     .     (1)  a'x^  +  h'x-^c'  =  0    .     .     (2) 

multiplying  the  first  by  a',  the  second  by  a,  and  subtracting, 
Ave  get 

(o?y')x  +  («c')  =  0 (3) 


where  (ah')  = 


a      b 


and  (etc')  = 


",    ^,    [See  Art.  17,  (3)], 


and,  again,  multiplying  the  first  by  c',  the  second  by  c,  sub- 
tracting, and  dividing  by  x,  we  get 

(ac>  +  (6c')  =  0 (4) 

The  problem  is  now  reduced  to  elimination  between  two 
linear  equations,  and  the  result  is 

{ac')-+{ba'){bc')  =  0 (5) 

This  method  of  forming  the  resultant  is  practically  very 
limited  in  application,  as  it  becomes  very  tedious  for  equations 
higher  than  the  fourth  degree. 

122.  Euler's  Method  of  Elimination.  Having  given  two 
equations  of  the  Hith  and  ?ttli  degrees  respectively, 

F{x)  =  b^'^  +  b,x"-'  +  ...  +  Z;,.  =  0  j     *     *     '     ^' 

we  propose  to  eliminate  .r,  or  to  find  their  resultant. 

If  these  equations  admit  a  common  root  r,  we  may  assume 

f{x)  =  {x-r)f,{x), 

F{x)  =  {x-r)F,{x\ 

/,(.f)  =  «,.c"'-'  +  «2.r"--  +  . •  •  +  «,„i  I  ,^. 

^^'''  i^.(.r)^;8,.-  +  ^^-+-  +  ^J     •     •     •     ^-^ 

the  coefficients  being  undetermined  quantities  depending  on  r. 


168 


THEORY  OF  EQUATIONS. 


Art.  122 


"Whence  we  have 

an  identical  equation  of  the  (m  -\-n  —  l)th  degree.  Now, 
equating  the  coefficients  of  like  powers  of  x  on  both  sides  of 
the  equation,  we  have  m  +  n  homogeneous  equations  of  the 
first  degree  in  the  m  +  n  quantities  Jh,  Ih,  '••  Pm,  Qi,  q->,  •••  Qn', 
and  eliminating  these  quantities  by  the  method  of  Art.  44,  we 
obtain  the  resultant  of  the  two  given  equations  in  the  form  of 
a  determinant.  The  method  will  be  made  clear  by  a  few  ex- 
amples. 

EXAMPLES. 
1.    Find  the  resultant  of  the  two  equations 

ax-  +  bx  -j-c  —  0,     GiX^  +  bjX  -f  Cj  =  0, 
supposing  thein  to  have  a  common  root.     We  have  identically 

(q^x  +  (h)  (ax-  +  bx  +  c)  =  (pix  +2h)  (ckx^  +  biX  +  Cy), 
or         (<?!«  ^  i'i«i)  -^^  +  (^1^  +  gM  -  pi&i  -  2h(ii)  ^ 

+  (9iC  +  q-:b  —  IhCi  -  P-A)  -c  +  g^c  -  iXCi  =  0. 

Equating  to  zero  all  the  coefficients  of  this  equation,  we 
have  the  four  homogeneous  equations 

g^a  —  i>i«i  =  0, 

gi^  +  Q'2<*  —  Pi^i  —  P2«i  =  0) 

g^c  +  g^b  —  PiCi  —2hh  =  0, 

gaC  —  P2C1  =  0, 

and,  eliminating  pi,  jh^  g^  %  "^^e  obtain  the  resultant  in  the 
form 


a 

0 

«i 

0 

b 

a 

&i 

a, 

c 

b 

Ci 

bi 

0 

c 

0 

Cl 

Art.  1-23 


ELIMIXA  TIOX. 


Kj'J 


The  student  can  easily  verify  that  this  result  is  the  same  as 
that  of  Art.  121. 

2.    Find  the  resultant  of  the  equations 

a^  +  «i-<r  +  ci.>x  +  «3  =  0,     6o.r  +  biX  +  62  =  0. 
Euler's  identity 

(«oX^  +  ctyV-  -f  a.^  +  cis)  (ftox  +  /?i) 
—  (b(ix^  +  b^x  +  b.;)  («oX-^  +  UiX  +  ag)  =  0, 

gives  the  following  five  equations : 

«o,5o  —  ^i«0  =  0, 

Oi^O  —  Oo/?1  —  ^i«o  —  ^u«i  =  0, 

«3i3o  +  Oji^i  —  b.Ui  —  Z*i«..,  =  0, 

a-ifSi  —  b-^Ui  =  0 ; 

whence 


22  = 


123.    Sylvester's  Dialytic  Method  of  Elimination.    This  method 

leads  to  the  same  determinants  for  resultants  as  Euler's  method ; 
but  it  is  simpler  in  its  application  and  has  an  advantage  over 
Euler's  method  in  point  of  generality,  sinc^e  it  can  often  be 
applied  to  form  the  resultant  of  equations  involving  several 
variables. 

To  find  the  resultant  of  the  two  equations 

f(x)  =  ciox""  +  a^x^-'  +  a.^^-'-  +  •••  +  a«  =  0, 

F{x)  =  ftoOf  +  &,x"-^  +  6.-C"-'  +  •..  +  i„  =  0, 


«o     0 

-bo 

0 

0 

tti-ttu 

-bi 

-K 

0 

a-i  -Oi 

-h 

-b, 

-bo 

a^    cu 

0 

-b. 

-61 

a     eta 

0 

0 

-b. 

170  THEORY  OF  EQUATIONS.  Art.  123 

of  degrees  m  and  n,  with  one  nnknown,  we  multiply  the  first 
successively  by 

U/,      *^.l      '^J      "^J       ***?  3 

and  the  second  by     aP,  «',  .t^,  or',  •••,  a;"'"^ 

We  obtain  thus  the  system  of  equations 

fix)  =  0,    xf(x)  =  0,    :v^(x)  =  0, . . .  .x"-y (a-)  =  0, 

J'(a-)  =  0,  xF(x)  =  0,  x-F{x)  =  0,  .••  x^^-^F{x)  =  0. 

There  are  m  +  «  equations,  and  the  highest  power  of  x  is 

m  +  ?i  —  1. 

If  there  is  a  common  root,  it  will  satisfy  all  the  equations 
of  this  system.  And,  in  taking  for  unknowns,  the  different 
powers  of  x, 

the  preceding  equations  form  a  system  of  m  -f-  n  linear  equa- 
tions with  m  -\-n  —  \  unknowns. 

Hence,  by  Art.  43,  we  can  eliminate  these  unknowns  and 
get  a  resultant,  i?,  which  is  equal  to  zero,  if  the  equations 
are  consistent. 

EXAMPLES. 

1.    Find  the  resultant  U  of  two  quadratic  equations 

ax-  -f  &.»  -f-  c  =  0,  Oio;-  +  \x  -f  Cj  =  0. 

We  have         xfix)  =    ax^  -\-  hx?  +  ex         —  0, 

/(.!•)=  a:^\hx^c  =0,    ■ 

xF{x)  =  ciiX^  +  b^x-  +  c^x         =  0, 

F(x)  =  a,x^  +  b^x  +  Ci  =  0 ; 

from  which,  eliminating  .r"*,  ai',  x,  we  get  the  same  determinant 
as  in  the  preceding  article,  columns  now  replacing  rows : 


Art.  124 


ELIMlXAriON. 


171 


a 

b 

c 

0 

0 

a 

b 

c 

«i 

^ 

t'l 

0 

0 

"i 

^ 

Ci 

R 


2.   Find  the  resultant  of  the  two  equations 

/(.r)  =  a^^x*  +  Oior'  +  "l-*-''  +  "y^  +  04  =  0, 
F(a-)  =  60^-'  +  6iX-  +  b,  =  0. 

We  have  the  following  system : 

f{x)  =  0  •x^  +  ttox*  +  GiX^  +  o,,.r  +  a^x  +  a*  =  0, 
SK/(a;)  =  (tox^  +  ciix*  +  cuv"  +  a^x^  +  «4^  +  0  =  0, 
F(x)  =  0-x^  +  0'x'4-0'X^+boX^+  bix  +  b.,  =  0, 
xF(x)  =  0  •  af  +  0  .  x'  +  ^o-^''  +  ?>!•'>-'  +  ^2^^'  +  0  =  0, 
x^F(x)  =  0  .  of  +  M'  +  ^i-^''  +  b-^x^  +  0  .  a-  +  0  =  0, 
x^F(^x)  =  b^py'  +  Z^i-x-*  +  ft.!'"'  +  0  •  .x-2  +  0  •  a;  +  0  =  0. 
Therefore,  we  have  for  the  resultant, 

0  ffo  «!  Oo  ttg  a^ 

f'o  f^i  f'2  f's  «4  0 

0  0  0  &o  ^1  ^2 

■^^0  0  60  ^  ^2  0 

0  bo  b,  60  0  0 

bo  b,  b,  0  0  0 


124.  There  are  other  methods  of  elimination,  notably  the 
nu'lhud  by  Symmetric  Functions  and  Bezout's  Method,  for  an 
explanation  of  which  we  refer  the  student  to  a  higher  work  on 
the  subject,  such  as  Burnside  and  Pantou's  Theory  of  E'juatious. 


172  THEORY  OF  EQUATIONS.  Art.  124 

We  shall  close  this  chapter  by  giving  some  examples  illustra- 
tive of  the  methods  that  we  have  considered  in  the  foregoing 
articles. 

EXAMPLES. 

Ic  Eliminate,  by  the  method  of  Art.  122,  x  from  the  two 
qiiadratic  equations 

0?  -I-  4  ;f  -  21  =  0,   ar  -  13  x'  +  30  =  0, 

and  show  that  R  =  0,  and  thus  prove  that  the  equations  have 
a  common  factor. 

2.  Apply  the  same  method  to  find  the  resultant  of  the  two 
cubic  equations 

ax^  +  bx^  +  ex  4-  d  =  0, 

a'x^  +  b'x-  +  c'x  +  cV  =  0. 

3.  To  solve,  making  use  of  Euler's  method,  the  equations : 

3  2/2  +  4  xy  ^3x^  —  9y-15  x  =  0 


f'-2xy  +  x^-\-2y-10x==0.}      •     '     '     '     (^) 

Rearranging  the  terms  according  to  descending  powers  of  x, 
e  have 


3 x^  +  (4.y  -  15)x  +  3 y-  -  9 y  =  0, 
x'-(10  +  2y)x  +  f-  +  2y  =  0. 


}     ....     (2) 


These  are  equations  of  the  second  degree  with  respect  to  x, 
of  which  the  coefficients  a,  b,  c,  Ui,  b^,  Cj  (see  Art.  122,  Ex.  (1)) 
are  respectively 

3,  4?/ -15,  3f--9y; 

1,  -(10  +  2./),  f  +  'iy. 

Therefore,  by  substitution  in  the  value  of  R  of  Ex.  (1),  Art, 
123,  we  have 


Vit.  124 

ELIMIXATION. 

3 

0                         10 

4?/ -15 

3            _(10  +  2v/)              1 

oy--9y 

4I/-15         r  +  2^         -(10  +  27/) 

0 

3r-9y              0                   ^f  +  2y 

173 


=  0  .     (3) 


or,  ill  developing, 

//?^y(y'  +  2y^-9y-18)  =  0     ....     (4) 

The  solution  of  this  equation  gives  for  the  roots 

y  =  0,  y=?,,  y  =  -3,  y  =  -2. 

Then,  to  calculate  the  corresponding  values  of  x,  in  this 
example,  we  simply  eliminate  ar  between  the  proposed  e<iua- 
tions,  which  gives  an  equation  of  the  tirst  degree  in  x 

(3  +  2?/).f-3^  =  0. 

Substituting  successively  the  roots  obtained  for  y,  we  find 

X  =  0,  x  =  \,  X  =  3,  x  =  6. 

The  given  equations  admit,  therefore,  four  common  solutions 

(0,0),  (3,1),  (-3,3),  (-2,6). 

Q  ^  ^.   Find  the  condition  that  all  the  roots  of  the  equation 

x'-\-ZHx  +  G  =  () 
shall  be  real. 

Solution.     The  three  roots  may  be  represented  by 

«,  ^+V?,  and  (3-V?. 

These  will  all  be  real  when 

/•^>0 (1) 

and  the  last  two  will  be  imaginary  when 

r<0 >-; 


174 


THEORY  OF  EQUATIONS. 


Art.  124 


Now  we  have,  Art.  94,  (3), 

2af3  +  (3'-y'  =  3H, (3) 

«^'  —  ay-  =  —  G. 

To  eliminate  a  and  (3  from  (3),  we  substitute  the  value  of  a 
from  the  iirst  in  the  second  and  third,  and  then  multiply  the 
second  by  (3  twice,  and  the  third  by  /3  once,  thus  forming  the 
hve  equations: 

3^-  +  (r  +  Si7)=0, 

3fi'  +  (f+'SH)(3  =  0, 

3{S*  +  iy'  +  3H)(3'  =  0, 

2(3'-2y'l3-G  =  0, 

2  ;8^  -  2  y'/B'  -G(3^0, 

whence,  the  determinant 


0  3 

3  0 

0     (y'  +  3H) 
0 

0  -2y2 

This  reduces  to 

72 


0       2 


0  (y'  +  SII) 

(y'  +  3II)  0 


0 


27    G-  +  4H^ 


0 


G 


0. 


4    (iy'  +  dliy 

which,  compared  with  (1),  shows  that  the  roots  are  all  real  when 

G'  +  4H'<0, 
the  required  condition.     When 

G'  +  4  W  >  0, 

the  two  conjugate  roots  are  imaginary.     The  function  Gr^+4iZ'^ 
is  called  the  dhcriminant  of  the  cubic  — ' 

x^  +  3  IIx  +(7  =  0. 


CHAPTER   XI. 

SOLUTION   OF   NUMERICAL   EQUATIONS. 

125.  Tliere  is  an  essential  difference  between  the  solutions 
of  algebraic  and  numerical  equations.  In  the  former  we  have 
a  general  result  expressed  in  symbolic  characters,  and  it  has 
been  proved  to  be  impossible  to  carry  this  solution  beyond 
equations  of  the  fourth  degree  (Art.  53). 

But  it  is  possible  to  solve  numerical  equations  of  a  much 
higher  degree,  and  to  obtain  at  least  appi'oximate  values  of  the 
roots  accurate  enough  for  all  practical  purposes. 

To  this  end,  we  determine  the  roots  separately,  and  Ave  must 
first  separate  the  roots ;  for,  before  attempting  the  approxima- 
tion to  any  individual  root,  it  is  generally  necessary  that  it 
should  be  situated  in  a  known  interval  which  contains  no 
other  real  root.  In  Chapter  IX.  certain  methods  of  separating 
the  roots  of  an  equation  have  been  explained. 

Real  roots  of  numerical  equations  are  either  commensurable 
or  incommensurable.  Commensurable  roots  include  integers, 
fractions,  and  repeating  decimals  which  can  be  reduced  to 
fractions;  incommensurable  roots  consist  of  interminable  deci- 
mals. The  roots  of  the  former  class  can  be  found  exactly,  and 
those  of  the  latter,  as  we  have  just  intimated,  approximated 
to  with  any  degree  of  accuracy.  In  this  chapter  we  shall  con- 
sider the  solution  of  numerical  equations. 

126.  Theorem.  If  the  coefficient  of  the  first  term  of  f(x)  is 
unitii  and  all  the  other  coefficients  are  irhole  numbers,  an;/  com- 
mensurable real  roof  of  f(x)  =  0  is  a  whole  number  anil  an  exact 
divisor  of  p„. 

176 


176  THEORY  OF  EQUATIONS.  Art.  120 

For,  if  possible,  let  ~,  a  fraction  in  its  lowest  terms,  be  a 
root  of  the  equation 

f{x)  =  X''  +2hx''-'^  -\-2hx'"'^  +  •••  +2^n-ix  +Pn  =  0; 
we  have  then 

from  which,  multiplying  by  6""\  we  obtain 

h 

Now,  since  —  is  a  fraction  in  its  lowest  terms,  this  equation 
h 
is   impossible,  for  an  integer  cannot  be  equal  to  a  fraction. 

Hence  -  cannot  be  a  root  of  the  equation.     The  real  roots  of 

b 
the  equation,  therefore,  are  either  integers  or  incommensurable 
quantities. 

It  is  evident,  by  Art.  94,  that  'any  commensurable  root  is  an 
exact  divisor  of  2^,i-  Every  equation  with  finite  coefficients 
can  be  reduced  to  the  form  in  which  the  coefficient  of  the  first 
term  is  unity,  and  those  of  the  other  terms  whole  numbers  by 
the  method  of  Art.  103. 


127.  Knowing  that  the  integral  roots  of  f{x)  are  factors  of 
p,„  we  can  often  determine  them  by  trial.  To  do  this,  we  must 
first  find  the  limits  within  which  the  roots  lie  (Chap.  IX). 
For  example,  take  the  equation 

af»_4.T2  +  a;  +  6  =  0. 

Here  the  real  roots  lie  between  +  4  and  —  2.  The  possible 
commensural)le  roots,  being  integral  factors  of  6,  are  ±1,  +  ?. 
+  3,  and  we  easily  find  that  the  roots  are  —  1,  +2,  -|-  3. 


Art.  128  NUMERICAL    EQUATIONS.  177 

We  shall  in  the  next  article  explain  a  general  method  of 
obtaining  the  integral  roots  of  an  equation  whose  coellicieuts 
are  all  integers. 

128.   Newton's  Method  of  Divisors. 

Suppose  h  to  be  an  integral  root  of  the  equation 

((o-r" -^  ciiX"  '+..•+ a„_i.T  +  o„  =  0.     .     .     .     (1) 

Let  the  quotient,  when  the  polynomial  is  divided  by  x  —  h,  be 

b^i"'-^  +  ^i.f"-''  +  •••  +  6,._2.^•  +  6„_i, 

in  which  h^,  hi,  etc.,  are  all  integers. 
Proceeding  as  in  Art.  82,  we  obtain 

Oo  =  b^„  rtj  =  &j  —  /<&„,  a.^  —  ho  —  hbi  ••• 

a„_2  =  6„_2  —  '*^n-3>    C*,.-!  =  ^n-l  —  ^*^-.-2>    «,.  =  — '*^h-1' 

The  last  of  these  equations  proves  that  «„  is  divisible  by  h, 
the  quotient  being  —  b„_i.  The  second  last,  which  is  the 
same  as 

a,._i  +  ^  =  -//6,._,, 
h 

proves  that  the  sum  of  the  quotient  thus  obtained  and  the 
second .  last  coefficient  is  again  divisible  by  /*,  the  quotient 
being  —  b„_2 ;  and  so  on.  Continuing  the  process,  the  last 
quotient  obtained   in  this  way  will  be  —  b^,  which  is  equal 

to  —  tto- 

In  this  way  we  can  test  all  the  divisors  of  a„  and  see 
Avhether  they  are  roots  of  the  equation.  They  must,  at  each 
step  of  the  above  process,  give  integral  quotients  and  a  final 
quotient  equal  to  —  a^.  As  soon  as  a  fractional  quotient  is 
met  with,  the  number  that  we  are  trying  must  be  rejected,  for 
it  cannot  be  an  integral  root.  This  is  called  Newton's* 
Method  of  Divisors. 


178  THEORY  OF  EQUATIONS.  Art.  129 

129.  Application  of  the  Method  of  Divisors.  In  applying 
this  method  it  is  convenient,  after  a  manner  analogous  to 
Art.  82,  to  write  the  series  of  operations  as  follows : 

a„         a„_i  a„_2  •••     ag  «!         a,, 

—  b„_i       —  6„_2       —  ^2       —  ^1     —  *o 


—  hi>„_2     —  hb„_3     —  hbi     —  hbo         0 

The  first  figure  in  the  second  line  (—  6„_i)  is  obtained  by- 
dividing  a„  by  h.  This  is  to  be  added  to  a„_i  to  obtain  the 
first  figure  in  the  third  line  (—  hb„_2).  This  is  to  be  divided 
by  h  to  obtain  the  second  figure  in  the  second  line  (— 6„_2); 
this  to  be  added  to  a„_2,  and  so  on.  If  h  be  a  root,  the  last 
figure  in  the  second  line  thus  obtained  will  be  —  cIq. 

AVhen  we  have  proved  in  this  manner  that  h  is  a  root,  the 
next  operation  with  any  divisor  may  be  performed,  not  on  the 
original  coefficients  a„,  a„_i,  •••,  but  on  those  of  the  second  line 
Avith  their  signs  changed,  for  these  are  the  coefficients  of  the 
quotient  when  the  original  polynomial  is  divided  by  x  —  h. 

We  need  not  include  the  numbers  1  and  —  1  in  the  number 
of  trial  divisors.  It  is  more  convenient  to  determine  before- 
hand by  trial  whether  either  of  these  numbers  is  a  root. 


EXAMPLES, 

1.   Find  the  integral  roots  of  the  equation 

x^  +  6  r^  +  X-  -  24  X  -  20  =  0. 

We  observe  that  all   the  roots  lie  between   +  3  and   —  6. 
Hence,  the  following  divisors  of  20  are  possible  roots: 

-5,     -4,     -2,     -1,     +1,     +2. 

By  trial  we  find  that  —  1  is  a  root,  and  + 1  is  not. 
We  commence  with  +  2. 


Alt.  129  NUMERICAL   KQUATIONS.  ITD 

_  20     -24     +1       +6+1 
_  10     -17     -8     -1 

Hence  2  is  a  root. 

We  next  try  —  2,  making  use  of  the  coefficients  of  the  second 
line  with  the  sign  changed. 

10         17         S         1 
-    5     -()     -1 

+  12     +^     ~0 
Hence  —  2  is  a  root. 

We  proceed  next  with  —  4.  As  this  does  not  divide  5,  it  is 
not  a  root,  so  we  try  —  5. 

5         6         1 
-1     -1 

""5     ~~0 
and  —  5  is  a  root. 

One  step  more  in  the  process  would  show  ns,  as  we  already 
know,  that  —  1  is  also  a  root.  Hence  the  roots  of  the  equa- 
tion are  —  1,  —  2,  —  5,  2. 

2.    Find  the  integral  roots  of  the  equation 

x'  +  11  .r  +  41  x""  +  ()1  X  +  30  =  0. 

It  is  evident  that  there  is  no  positive  root.  By  trial  we 
find  that  the  limit  of  the  negative  roots  is  —  6.  Hence  the 
possible  integral  roots  are 

-1,     -2,     -3,     -4,     -5. 

We  commence  with  —  5. 


30 

61 

41 

11 

1 

-    6 

-11 

-    6 

-1 

55 

30 

5 

0 

is  a  root. 

180  THEORY  OF  EQUATIONS.  Art.  129 

As  4  will  not  divide  6,  —  4  is  not  a  root  (as  we  knew  in  the 
beginning,  for  it  does  not  divide  oO),  so  we  try  —  3,  and  then 
—  2,  and  lastly  —  1,  as  follows : 

6         11         6         1 
-2-3-1 

9    ~3    ~0 
Hence,  —  3  is  a  root. 

2         3         1 
-1     -1 
2         0 
Hence,  —  2  is  a  root. 

1         1 

-1 

0 

Hence,  —  1  is  a  root,  and  the  roots  are  all  integral. 

3.  Find  the  integral  roots  of 

af  -4x*-  16  x-s  +  46  x'  +  63  x  -  90  =  0 

By  trial  we  find  that  +  1  is  a  root ;  we  therefore  depress 
the  equation  by  dividing  through  by  a;  —  1,  which  gives 

x'  -  3  x"  -  19  .T-  +  27  a;  +  90  =  0. 

Ans.  1,  3,  5,  -  2,  -  3. 

4.  Find  all  the  roots  of 

X*  -Sx"-  11  x""  +  19  a;  +  42  =  0. 

Here  limits  of  the  roots  are  +  4  and  —  3 ;  and  the  possible 
integral  roots  are  +3,  +2,  +1,  —  1,  —  2. 

Ans.   +  3,  -  2,  1  +  2  V2,  1  -  2  V2. 

5.  Find  all  the  roots  of  the  equation 

x'  +  ar^  -  2  a;2  ^  4  a;  -  24  =  0. 

6.  Find  the  integral  roots  of  the  equation 

15  .^•^  -  19  x'  +  6  x^  +  15  x-  -  19  .r  +  6  =  0. 


Art.  131  NUMERICAL   EQUATIONS.  181 

7.    Fiud  all  the  roots  of  the  equation 

x*-2  x'  -  19  X"  +  68  a;  -  60  =  0. 

The  roots  lie  between  —6  and  6.  We  find  that  2,  3,  —5 
ai-e  roots,  and  that  the  factor  left  after  the  final  division  is 
a-  —  2 ;  hence  2  is  a  double  root,  and  the  polynomial  is  there- 
fore equivalent  to 

(.^•-2j^•-3)(.^•  +  5). 

130.  Determination  of  Multiple  Roots.  The  j\retliod  of 
Divisors,  as  shown  by  Ex.  7  of  the  last  article,  determines 
multiple >oots  when  they  are  commensurable.  In  applying 
the  method,  Avdien  any  divisor  of  a,„  which  is  found  to  be  a 
root,  is  a  divisor  of  the  absolute  term  of  the  reduced  poly- 
nomial, it  may  also  be  a  root  of  the  latter.  If  it  is,  it  will  be 
a  double  root  of  the  proposed  equation.  If  it  is  found  to  be 
a  root  of  the  next  reduced  polynomial,  it  will  be  a  triple  root 
of  the  proposed  equation,  and  so  on.  It  is  often  a  saving  of 
labor  to  seek  for  multiple  roots  in  this  way,  rather  than  by  the 
laborious  method  of  the  H.  C.  F.  (Art.  98). 

EXAM  PLES. 
Find  the  commensurable  and  multiple  roots  of 

1 .  2  x"  -  31  .t--  +  1 12  .i-  -h  04  =  0. 

2.  x'  -  x"  -  30  .«■-  -  76  X  -  50  =  0. 

3.  .c'  _  8  x'  +  22  .*;■'  -  20  x-  +  21  .t-  -  18  =  0. 

131.  Newton's  Method  of  Approximation.  AVe  shall  now 
proceed  to  the  determination  of  incommensurable  roots,  giv- 
ing iirst  Newton's  Method. 

In  any  method  of  approximation,  the  root  that  we  are  seek- 
ing is  supposed  to  be  separated  from  all  other  roots  and  to  be 
contained   within   close   limits.      Let  f(x)  =  0   be   the   given 


182  THEORY  OF  EQUATIONS.  Art.  131 

equation,  and  let  a  be  a  known  number  differing  by  a  small 
quantity  (a  decimal  fraction),  h,  say,  from  the  root  a  -f  h. 
We  have  then 

f(a  +  h)=f(a)+f'{a)h+'^^h'+-=0..     .     (1) 

In  the  first  approximation,  since  h  is  small,  we  neglect  the 
terms  which  contain  /r  and  higher  powers. 
Hence  (1)  becomes 

/(a)  +f(a)h  =  0, 

which  gives,  as  a  first  approximation  to  the  root,  the  value 

Kepresenting  this  root  by  b,  and  applying  the  same  process 
a  second  time,  we  have  for  a  second  approximation  to  the  root 

b  —  ^Wj  and  so  on. 

The  oftener  this  process  is  repeated,  the  more  accurate  is  the 
approximation.  In  general  the  approximation  is  rapid,  but 
this  method  has  been  entirely  superseded  by  Horner's  IMethod, 
which  we  take  up  in  the  next  article.  To  illustrate  this 
method,  consider  the  equation 

/(.T)  =  .r^-4.^'2-2.^•  +  4  =  0 (1) 

We  find  that  the  three  roots  are  comprised  respectively  in 
the  intervals  (- 1,  -  2),  (0,  1),  (4,  5).  Let  us  first  calculate 
the  last  one.  Narrowing  the  limits,  we  find  that  the  root  is 
comprised  between  4.2  and  4.3. 

We  find,  then, 

/(g)  _/(4.2)_- 0.872  _ 

/'(a)  "/'('i.2) -+17.32-      ^•^''^^- 


Art.  132  NUMERICAL   EQUATIONS.  183 

A  first  approximation  is,  therefore,  4.2504. 
Calling  this  b,  we  have 

f(b)  _  ./•(4-2504)  _  0.014825 
fW)  -/'(4.2504)  -  18.1840  "O-OOOSl^- 

A  second  approximation  is 

4.2504  -  0.000815  =  4.249585, 

which  will  be  found  to  be  correct  to  the  third  decimal  place. 
In  like  manner  the  root  between  0  and  1  is  found  to  be  0.853G3, 
and  that  between  —1  and  —2,  to  be  —1.102775.  Here,  as 
the  example  is  given  simply  to  illustrate  the  method,  no  pains 
has  been  taken  to  carry  the  approximation  beyond  the  third 
decimal  place.  As  the  coefficient  of  x-  in  equation  (1)  is  4,  the 
3  roots  added  together  should  give  4. 

132.   Homer's  Method  of   Solving  Numerical  Equations.     By 

Horner's  method  both  the  commensurable  and  the  incom- 
mensurable roots  can  be  obtained.  The  root  is  evolved  tigure 
by  figure ;  first  the  integral  part  (if  any),  then  the  decimal  part 
till  the  root  terminates  if  commensurable,  or  to  any  number  of 
places  if  incommensurable.  This  inethod  is  really  an  extension 
of  the  principles  of  the  method  of  Art.  107,  which  involves  the 
diminishing  of  the  roots  by  known  quantities.  A  X'oot  Avliich 
has  several  figures  is  obtained  by  continued  applications  of 
that  method,  the  successive  transformations  being  exliibitod  in 
a  compact  form,  as  will  be  made  apparent  by  the  examples 
given  below. 

The  first  step  in  the  solution  of  a  numerical  equation  is  to 
find  the  Jirst  figure  of  the  root.  This  can  usually  be  done  In- 
trial,  though  sometimes  it  may  be  necessary  to  resort  to  one  of 
the  methods  of  Chapter  IX  to  separate  the  roots. 


184 


THEORY  OF  EQUATIONS. 


Art.  132 


EXAMPLES. 
1.    Find  the  positive  roots  of  tlie  equation 

8  .r*  -  260  .^■2  -  546  x  -  207  =  0. 

There  can  be  only  one  positive  root ;  and  it  is  found  by  trial 
to  lie  between  30  and  40.  Thus  the  first  figure  of  the  root  is 
3.  We  now  diminish  the  roots  by  30.  The  transformed  equar 
tion  will  have  one  root  between  0  and  10.  It  is  found  to  lie 
between  4  and  5.  We  next  diminish  the  roots  of  the  trans- 
formed equation  by  4,  so  that  the  roots  of  the  proposed 
equation  will  be  diminished  by  34.  The  second  transformed 
equation  will  have  one  root  between  0  and  1.  On  diminishing 
the  roots  of  this  latter  equation  by  .5,  we  find  that  its  absolute 
term  is  reduced  to  zero ;  that  is,  the  diminution  of  the  roots  of 
the  proposed  equation  by  34.5  reduces  its  absolute  term  to 
zero. 

Therefore,  34.5  is  a  root  of  the  given  equation.  The  method 
of  calculation  is  exhibited  as  follows  : 


8 


-260 

-  546 

-  207  1  34.5 

240 

-  600 

-  34380 

-  20 

-1146 

-  34587 

240      6600 

29688 

220 

5454 

-4899 

240 

1968 

4899 

460      7422 

0 

32      2096 

492 

9518 

32 

280 

524 

9798 

32 

556 

4 

660 


Art.  132 


NUMERICAL    EQUATIONS. 


185 


The  broken  lines  mark  the  conclusion  of  each  transforma- 
tion, and  the  ligures  in  dark  type  are  the  coefficients  of  the 
successive  transformed  equations.     (See  Art.  107.) 

Thus  8  x"  +  4G0  x"  +  5454  x  -  34587  =  0 

is  the  first  transformed  ec^uation,  whose  roots  are  less  by  30 
than  the  roots  of  the  proposed  equation,  and  are  found  to  lie 
between  4  and  5.     And 

8  r*  +  556  x^  +  9518  x  -  4899  =  0 
is  the  second  transformed  equation. 

If  this  second  transformed  equation  had  not  an  exact  root 
.5,  we  should  find  the  limits  between  which  the  root  lies,  and 
then  proceed  as  before,  and  so  on. 


2.    Find  the  positive  root  of  the  equation 

4ar'-13a;2-31cc-275  =  0.      . 
Here  the  arithmetical  calculation  is  as  follows ; 


(1) 


-13 

24 

~11 
24 


-31 
66 

~35 

210 


6.25 


210 


65 

51.392 


35 

245 

-13.608 

24 

11.96 

13.608 

59 

256.96 

0 

.8 

12.12 

69.8 
.8 

269.08 

3.08 

60.6 

272.16 

.8 

61.4 
.2 


186  THEORY  OF  EQUATIONS.  Art.  132 

We  find  by  trial  that  the  proposed  equation  has  its  positive 
root  between  6  and  7.  The  first  figure  of  the  root  is,  there- 
fore, 6. 

Diminish  the  roots  by  6.     The  transformed  equation 

4  a^  +  59  ar  +  245  a;  -  65  =  0 

has  a  root  between  0  and  1 .  It  is  found  by  trial  to  lie  between 
.2  and  .3. 

Diminish  the  roots  again  by  .2.  The  transformed  equa- 
tion 

4  x3  4_  61.4  r"  +  269.08  x  -  13.608  =  0 

is  found  to  have  the  root  .05.  Hence  6.25  is  a  root  of  the 
proposed  equation. 

It  is  convenient  in  practice  to  avoid  the  use  of  the  decimal 
points.     This  can  easily  be  effected  as  follows : 

When  the  decimal  part  of  the  root  (suppose  .abc  •••)  is  about 
to  appear,  multiply  the  roots  of  the  corresponding  transformed 
equation  by  10;  that  is,  annex  one  zero  to  the  right  of  the 
figure  in  the  first  column,  two  to  the  right  of  the  figure 
in  the  second  column,  three  to  the  right  of  that  in  the  third ; 
and  so  on,  if  there  be  more  columns  (Art.  103).  The  root 
of  the  transformed  equation  is  then,  not  .abc  •••,  but  a.bc  •••. 

Diminish  the  roots  by  a.  The  transformed  equation  has  a 
root  .be  ••■.  Multiply  the  roots  of  this  equation  again  by  10. 
The  root  becomes  b.c  •••,  and  the  process  is  continued  as 
before. 

To  illustrate  this  we  repeat  the  above  operation,  omitting 
the  decimal  points.  In  subsequent  examples  in  this  book 
this  simplification  will  be  adopted,  and  the  student  is  advised 
to  make  use  of  this  principle  in  the  solution  of  all  such 
examples. 


Art.  133 
4 


■13 
24 

11 

24 

35 
24 


NUMERICAL   EQUATIONS. 

-31 
6G 


187 


590 

8 

598 
8 

606 


35 
210 


24500 

111)6 

25696 
1212 


2690800 

30800 

2721600 


65000 

5l;;i)2 


-13608000 

136(18000 


6140 

20 

6160 


In  the  examples  here  considered  the  root  terminates  at  an 
early  stage.  When  there  are  many  more  figures  in  the  root, 
the  process  would  become  very  laborious,  if  it  were  not  for  a 
simplification  which  we  shall  explain  in  the  next  article.  This 
introduces  to  us  one  of  the  most  valuable  practical  advantages 
of  Horner's  Method,  which  is,  that  after  the  second  or  third 
(sometimes  even  after  the  first)  figure  of  the  root  is  found,  the 
transformed  equation  itself  sucjrjests,  by  mere  inspection,  the  next 
figure  of  the  root. 


133.   Principle  of  the  Trial-divisor.     We  have  seen  in  Art.  131 

that  when  an  e(|uation  is  transformed  by  the  substitution  of 
a  +  /<  for  a,  a  being  a  nundjer  dilforing  from  the  true  root  by  a 
quantity  h,  small  in  proportion  to  a,  an  approximate  numerical 


value  of  /i  is 


/(«_) 
/'(«)' 


188  THEORY  OF  EQUATIONS.  Art.  133 

Now,  as  in  the  successive  transformed  equations  of  Horner's 
method,  the  last  coefficient  is  /(a)  and  tlie  next  to  the  last  is 
f'(a),  we  would  evidently  get  the  next  figure  of  the  root  by- 
dividing  /(o)  by/'(o);  that  is,  by  dividing  the  last  coefficient 
by  the  coefficient  next  to  the  last.  This  will,  in  general,  give 
the  correct  figure  only  after  two  or  three  steps  in  the  process 
have  been  completed,  and  the  part  of  the  root  to  be  found 
bears  a  small  ratio  to  the  part  already  evolved.  We  might, 
therefore,  if  we  pleased,  at  any  stage  of  Horner's  operations, 
apply  Newton's  method  to  get  a  further  approximation  to  the 
root.  The  second  last  coefficient  of  each  transformed  equation 
is  called  the  trial-divisor.  It  is  evident  that  the  application  of 
this  principle  will  greatly  facilitate  the  work,  but  we  must  use 
due  care  not  to  apply  Newton's  method  too  soon. 

Thus,  in  the  second  example  of  the  last  article,  the  number 
5  is  correctly  suggested  by  the  trial-divisor  2690800,  for 
2090800  into  13608000  goes  5  times  (and  something  over,  of 
course).  In  this  example,  indeed,  the  second  figure  of  the 
root  is  correctly  suggested  by  the  trial-divisor  of  the  first 
transformed  equation ;  although,  in  general,  such  is  not  the 
case.  In  practice  the  student  must  estimate  the  probable 
effect  of  the  leading  coefficients  of  the  transformed  equation. 
To  illustrate,  consider  the  following  examples : 

EXAMPLES. 

1.   Find  the  roots  of  the  equation  a.*^  —  7  cc  +  7  =  0. 

We  first  separate  the  roots  by  Sturm's  Theorem  (Ex.  4, 
Art.  119).  We  find  that  there  are  two  positive  roots  between 
1  and  2,  and  a  negative  root  between  —  3  and  —  4.  Trans- 
forming the  equation  by  diminishing  the  roots  by  1,  we  find 
that  of  the  two  positive  roots,  one  lies  between  1.3  and  1.4, 
and  the  other  between  1.6  and  1.7. 

We  shall  find  the  first  root  to  five  deeimal  places,  and  leave 
it  as  an  exercise  for  the  student  to  find  the  root  between  1.6 
and  1.7  and  the  negative  root. 


I 


It.  l:};3 

NU 

MEltIC 

AL 

EqrATIONH.                           IHU 

The  calculation  is 

written  as  follows : 

0 

-7                               +7  1  1.35680 

1 

1 

-6 

1 

-6 

1000 

1 

-^  J 

-90;; 

2 

-400 

97000 

1 

1)9 
-301 

-  8(;(;i'5 

30 

10375000 

3 

108 

—  90  IS! IS  1 

33 

-19300 

1326016000 

3 

1975 
-173L>5 

-llSllL".).-,(iS 

3G 

141586432000 

3 

2000 

390 

-1532500 

o 
395 

24;5;!6 
-15(»S1(;4 

5 

2i;;72 

400 

-148379200 

r> 

3L>55()4 
-MS(r)360(; 

4050 

6 

',V2~^7A\)> 

4056 

-14772812800 

6 

4062 

6 

40680 

8 

40688 

8 
)6 

4061 

8 

407040 

Here  we  first  diminish  tlie  roots  by  1.  As  the  decimal  i»art 
is  about  to  appear,  attach  ciphers  to  the  coetKcieuts  of  the 
transformed  equation,  which  thus  becomes 


190  THEORY  OF  EQUATIONS.  Art.  133 

ips  +  30  X-  -  400  X  +  1000  =  0. 

We  next  diminish  the  roots  by  3,  as  we  have  already  found 
that  3  is  the  next  figure  of  the  root  souglit.  After  multiplying 
the  roots  by  10,  the  second  transformed  equation  is 

x"  +  390  X-  -  19300  X-  +  97000  =  0. 

The  trial  divisor  now  becomes  effective;  19300  into  97000 
goes  5  times,  and  5  is  found  to  be  the  next  figure  of  the  root. 
If  we  had  adopted  the  figure  6,  the  absolute  term  would  have 
become  negative,  the  change  of  sign  showing  that  we  had  gone 
beyond  the  root.  V\'e  must  take  care  that,  at  least  after 
the  first  transformation,  the  absolute  term  preserves  its  sign 
throughout  the  operation.  The  figure  to  be  adopted  in  every 
case  as  part  of  the  root  is  that  highest  number  which  in  the 
process  of  transformation  tcill  not  change  the  sign  of  the  absolute 
term.  If  Ave  were  to  take  by  mistake  a  number  too  small,  the 
error  would  show  itself,  just  as  in  ordinary  division  or  evolu- 
tion, by  the  next  suggested  number  being  greater  than  9. 

After  diminishing  by  5  the  roots  of  the  second  transformed 
equation  (and  multiplying  the  roots  of  the  resulting  equation 
by  10),  the  next  figure  of  the  root  is  6,  for  1532500  goes  into 
10375000  6  times.  And  so  we  proceed,  as  indicated  in  tlie 
above  operation.  Of  course  the  process  can  be  continued 
indefinitely,  and  the  root  obtained  correct  to  any  number  of 
decimal  places. 

2.  I^'ind,  to  5  decimal  places,  the  positive  root  of  the  equation 

which  lies  between  2  and  3. 

3.  Find  the  two  positive  roots  of  the  equation 

x'  +  4  ar''  -  4  a;2  -  11  X  +  4  =  0. 


Art.  l;5t  NUMERICAL   EQUATIONS.  191 

There  are  several  abbreviatiuns  of  Horner's  process,  by 
■which,  after  three  or  four  phices  of  decimals  have  been  calcu- 
lated as  above,  several  more  may  be  correctly  obtained  by  a 
contracted  process,  for  an  explanation  of  which  we  refer  the 

reader  to  Buruside  and  Panton's  77<co/v/  of  EtjmUiuits. 

134.  Negative  Roots.  To  obtain  a  negative  root  of  f(x)  =  0, 
we  simply  form  the  equation /(— a;)  =  0  (Art.  102),  and  get  its 
corresponding  positive  root,  which  will  be  the  required  negative 
root  of  f(x)  =  0. 

EXAMPLES. 

1.  Find  the  negative  root  of  the  equation 

a;*-12.^-  +  12x-3  =  0. 

2.  Find  the  root  between  3  and  4  of  the  equation 

X*  -  lb  .r  -  4 .1-  +  8  =  U, 
to  four  places  of  decimals. 

3.  Find  the  real  roots  of  the  equation 

iK*  -  12  .T  -f  7  =  0.      Ans.  2.0473 ;  .5937. 


MISCELLANEOUS   EXAMPLES. 
Find  the  quotient  and  remainder  when 

1.  x^  —  2  X'  +  3  X*  -  () .»;'  -  10  .»•  +  0  is  divided  by  x  -  2. 

2.  X*  +  3x^-2x-  +  x--i  is  divided  by  x  -  3. 

3.  4  x^  +  2  ar  4-  5  .c  -  9  is  divided  by  x  +  4. 

4.  2  x^  +  X*  -  4  .r'  +  8  ar  -  2  X  +  16  is  divided  by  a-  +  G. 

5.  a;"  +  2  .r"*  -  4  a;^  -  5  x"  +  ^  -  -p'  +  10  is  divided  by  x  -  5. 


192  THEORY   OF  EQUATIONS.  Art.  iJi 

6.  Trace  the  polynomial 

7.  Solve  the  equation 

a^  + 288  a; +  1216  =  0, 
which  has  a  root  2  —  10  V  —  3. 

8.  Form  a  rational  sextic  equation  which  shall  have  for 
three  of  its  roots 

1-3V2,    2+V^    3-2V^l. 

9.  Solve  the  cubic 

x"  +  100  x"  +  100  X  +  1000, 
one  root  being  V—  10. 

■Find  by  Descartes'  rule  an  inferior  limit  to  the  number  of 
imaginary  roots  of  the  following  equations : 

10.  x^-3  X'  +  x^  -  .T^  -  a:  +  6  =  0. 

11.  x'  +  4:x'''-\-x'-\-S  x'  +  x"  +  x'  +  x  +  l  =  0. 

12.  x'"  +  2  ;k^  -  5  x'  -  a;*  +  ar^  +  4  .^•2  -6  =  0. 

13.  x-«-5a;''4-3  =  0. 

14.  4.T''  +  7a,-^-18x-30  =  0. 

15.  Solve  the  equation 

27  x"  +  42  .^2  -  28  .T  -  8  =  0, 

whose  roots  are  in  geometric  progression.     [Art.  94.] 

Ans.   -  2,  f ,  -  f 

16.  The  equation 

X*  -  2  .t'^  +  4  a;2  +  6  .TJ  -  21  =  0, 
has  two  roots  equal  in  magnitude  and  opposite  in  sign;  deter- 
mine all  the  roots.     Take  a  +  /8  =  0,  and  use  Art.  94. 

Ans.  V3,  ^=^3,  1  ±  V^^. 


Art.  1:3 1  MISCELLAMiOUS   EXAMPLES.  l'J8 

17.  Oue  of  the  roots  of  the  cubic 

ix?  —  2^x-  +  qx  —  r  =  0 

is  double  another;  show  that  it  may  be  found  from  a  (juad- 
ratic  ecjuation. 

18.  Find    the    condition    which    must    be    satisiied    l)y    the 
coetticients  of  the  equation 

ar*  —  px^  +  qx  —  r  —  0, 

when  two  of  its  roots  a,  (3,  are  connected  by  a  rehation  a 4-/8  =  0. 

^l/(.s.  jiq  —  /•  =  0. 

19.  The  product  of  two  unequal  roots  of  the  equation 

ax^  +  b.i"  +  ex  +  <l  =  0 
is  1 ;  prove  that  the  third  root  is ^• 

Solve  the  fo'llowing  five  equations,  each  of  which  has  equal 
roots : 

20.  x"  -  5  ar'  -  8  .r  +  48  =  0. 

21.  .r<_|.r  +  ^  =  0. 

22.  x'  -  2  x"  -  a.-2  -  4  .^-  4- 12  =  0. 

23.  x'  +  2  x"  -  12  X-  -  18  X  +  27  =  0. 

24.  a;^  _  7  x«  +  10  x^  +  22  x'  -  43  x"  -  35  .r^  +  48  a;  -f  3r>  =  0. 

A».'i.  (x-2y(x-S)\x+lf. 

25.  Find  the  equation  wdiose  roots  are  the  roots  of 

a-'  -  6a,-«  -  X'  +  2  ar'  -  X  +  7  =  0 
with  their  signs  changed. 

26.  Change  the  equation 

2  .r*  _  4  x"  +  5  x"^  -  7  a-  +  3  =  0 
into  another,  the  coefficient  of   wliose  liighest  term  will    l>e 
unity,  and  the  coefficients  of  the  other  terms  integers. 


194  THEORY  OF  EQUATIONS.  Art.  134 

Remove  the  fractional  coefficients  from  the  equations : 

27.  x*-:^x'  +  ^a^-5x  +  2  =  0. 

28.  x^  +  ^jx'--j\^0. 

29.  x'  +  \x'-^x  +  3  =  0. 

30.  Find  the  equation  whose  roots  are  the  reciprocals  of  the 
roots  of 

x'  -  91  x~  -  910  x  +  1000  =  0. 

31.  Give  condition  that  the  following  equation  should  have, 
(1)  one  inhnite  root,  (2)  two  infinite  roots. 

(a^  -  4)  x^  +  (c  -  7)  x^  +  ax-  -  ex  +  20  =  0. 

32.  Increase  by  5  the  roots  of  the  equation 

2  x'  -  r'  +  6  .T-  +  3  .T  -  10  =  0. 

33.  Increase  by  3  the  roots  of  the  equation 

a^-3x^  +  x-7  =  0. 

34.  Diminish  by  2  the  roots  of  the  equation 

x^-x'^  +  x'^  —  x-{-5  =  0. 

35.  Diminish  by  6  the  roots  of  the  equation 

x'-3x^-2x  +  16  =  0. 

36.  Diminish  by  1  the  roots  of  the  equation 

x^-2x^  +  x^  +  4  ar'  -  a:*  +  7  .r-  -  16  =  0. 

37.  Increase  by  10  the  roots  of  the  equation 

3  x'  -Qx^-4x^  +  2x-5  =  0. 

Transform  each  of  the  following  three  equations  into  an- 
other  wanting  the  second  term : 

38.  x"  -  3  x"  +  4  .^•  -  4  =  0. 

39.  x^-8.r^  +  5  =  0. 


Art.  i;M  MISCELLA  XEO  Us  EX  A  Ml'L  KS.  1 :  t.j 

40.  2  x"  +  12  x'  -  3  .r  +  5  =  0. 

41.  Remove  the  tliird  term  in  the  equation 

x*-8  .r'  +  18  X-  -  15  X  +  14  =  U. 

Remove  the  secoml  term  and  solve  the  two  euhie  equations 
(Art.  110) : 

42.  x^  -  18  X-  +  157  X  -  510  =  0.  Aits.  6,  6  ±  T^AH". 

43.  x"  -  7  .^-'  +  11  -v  =  20.  .1,;.^.  5,  1  +  V^3, 1  -  V^^. 

Find  a  superior  limit  to  the  positive  and  negative  roots  of 
the  equations : 

44.  x'  -  5  x"  +  37  a.-2  -  3  .r  +  39  =  0. 

45.  CL^  +  7  a;*  -  12  x"  -  49  x-  +  52  x  -  13  =  0. 

Apply  Sturm's  theorem  to  determine  the  number  and  situa- 
tion of  the  real  roots  of  the  following  five  equations : 

46.  x*-4:X^  +  7  ar  -  6 .«  -  4  =  0. 

47.  X*  -  5  ar'  +  10  .«2  -  G  x  -21=0. 

48.  a.-*  -  10  x'  +  6x  +  l=  0. 

^l//.s'.  Roots  all  real;  one  in  the  interval  J  —  4,  —  3|;  two  in 
the  interval  |  — 1,  0|;  and  i)()sitive  roots  in  tlie  intervals  |0,  IJ, 
13,  4(. 

49.  x'  -  2  .r'  _  4  .i;  +  10  =  0. 

50.  x"  _  4  .^•2  -  4  a;  +  20  =  0. 

61.  Show,  by  Sturm's  tlieorem,  that  all  the  roots  of  the 
equation 

a^  -h  3  x^-jr-  3  x  +  H  =  <> 
are  imaginary. 


196  THEORY  OF  EQUATIONS.  Art.  134 

Find  the  integral  roots  of  the  following  equations : 

52.  X*  -  5  X'''  +  25 .7;  -  21  =  0. 

53.  9  x'  4-  30  x'  +  22  x'  +  10  0;=^  +  17  a;^  -  20  a;  +  4  =  0. 

54.  x'  +  6  r'  -  22  af  -  33  x-  +  54  =  0. 

55.  Find  the  commensurable  and  multiple  roots  of 

X*  +  12  a;3  +  32  a.-2  -  24  a-  +  4  =  0. 

A)is.  The    equation    has    two   pairs    of    equal   roots,  both 
incommensurable. 

56.  Find  the  commensurable  and  multiple  roots  of 

a;«  -  8  af  +  20  x'  -  32  x'  +  68  x-  -  32  a;  +  64  =  0. 

Ans.  (x  -  4)-  (if2  +  2)-  =  0. 

57.  Find,  by  Horner's  method,  to  six  decimal  places,  the 
root  between  2  and  3  of  the  equation 

ar^  _  49  X'  +  658  x  -  1379  =  0. 

58.  Find  the  two  real  roots  of  the  equation 

a;*  - 11727  »  + 40385  =  0.     Ans.  3.45592,  21.43067. 

Find  all  the  roots  of  the  three  equations : 

59.  x^+x--2x-l  =  {).    Am.  -1.80194,-0.44504,1.24698. 

60.  x^  -  315  x''  -  19684  x  +  2977260  =  0. 

61.  x''-10x'-\-C^x  +  l  =  0. 

-  3.065315791, 

-  0.691576280, 
Ans.   \  -0.175674799, 

+  0.879508708, 
+  3.053058162. 


APPENDIX  A. 

The  definitions  of  algebraic  and  transcendental  functions 
given  in  Art.  50,  page  78,  are  somewhat  broader  than  those 
found  in  our  elementary  text-books  on  Algebra.  That  these 
definitions  are  exact  and  cover  the  entire  ground,  is  evident 
from  the  following  considerations  : 

In  mathematics  there  are  only  four  fundamental  operations, 
namely :  addition,  subtraction,  multiplication,  and  division. 
If  two  quantities,  x  and  y,  are  so  related  that  when  one  of 
them  is  given  the  other  can  be  calculated,  the  one  is  said  to  be 
a  mathematical /»«c</oM  of  the  other.  Mathematical  functions 
are  further  divided  into  two  great  classes  according  as  the 
number  of  fundamental  operations  is  finite  or  infinite,  in  order 
to  calculate  the  function  when  the  other  quantity  is  given. 

//'  the  number  of  such  operations  is  finite,  the  function  is  said 
to  be  algebraic;  otherwise,  transcendental.     For  example;  in 


a^-5''  '  '  2jr  +  3x  +  r 

the  number  of  operations  on  x  is  Jinite.     Such  expressions  are 
algebraic  functions. 

The  student  familiar  with  trigonometry  will  recall  that  siux, 
e',  log  (1  +  x),  as  functions  of  x,  are ' 

sin  A-  =  a;  —    -  H ,  without  end, 

3!      5! 

e'  =  l-fa;+  —  -f-p-H ,  without  end, 

2 !      «3 ! 

log  (1  +  x)  =  X  —  J  r*  -f  !^  .1-^ ,  witliout  end. 

l'J7 


198  APPENDIX. 

These  are  examples  of  transcendental  functions,  for  tlie  num- 
ber of  operations  is  infinite.       -  

It  is  also  customary  to  consider  expressions  like  Va;^  +  2  x, 
\/^~+~«r+l  as  algebraic  functions,  although  to  derive  their 
true  value,  for  values  of  x  other  than  special  ones,  would  im- 
ply an  infinite  number  of  operations.  This  seeming  inconsist- 
ency may  be  explained  on  the  gi'ound  that  the  extraction  of 
the  root,  though  involving,  possibly,  an  infinite  number  of  the 
four  fundamental  operations,  is  counted  as  a  single  (though 
complex)  operation,  —  making  the  total  number  of  operations 
fiuite  (in  thought). 


APPENDIX   B. 

Argand's  Diagram:  In  Article  67  and  the  foot-note,  it  is 
possible  that  rather  too  much  credit  is  given  to  Argand. 

Kossak*  says  that  Kuhn,  in  Novi  Commentarii  Ara'l 
Petrop.  Ill,  ad  J 750-1731,  was  the  first  to  give  geometric 
expression  to  V—  1.     Thus  lay  off  OAo  —  1, 


CMi  =  — 1,  and   draw   the   perpendicular   OB  to   meet   semi- 
circle on  A^A^  at  B.     Then  OR  =  OA^  •  OA^, 

or  0R  =  -1.     .:   0B  =  V'^.1[ 


*  EUmente  der  Arithmetik. 

+  See  also  Cajori's  IliHta^if-afSBlth^matics,  page  317. 

199 


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